Are ESD safe tools necessary with proper setup?
When you say your tools are not ESD safe, do you mean they are all metal? As in all metal tweezers? Or that they have a metal working end but have plastic handles? Like screwdrivers and pliers? I would have no qualms about all metal tools since you are wearing the ESD wrist strap when you are grabbing the tool anyways.

Technically, all-metal tools aren't as safe as metal tools with dissipative handles which drain the charge in a slow and controlled manner. It mainly means that if you are charged for some reason (i.e. not wearing a strap) and you pick the tool up and touch it to a component, the charge will equalize slowly and not damage the component rather than quickly with a spark which would damage the component.

The more minor aspect is that if the metal tool itself is carrying a charge for some reason and isn't sitting on an ESD mat to drain the charge, or a component that is carrying a charge (maybe it wasn't on an ESD mat and was slowly able to accumulate charge so as not to damage it) and touches the metal tool, the rapidly charge equalization between the two will damage the component. Note the latter case can happen even if the metal tool is sitting on an ESD mat and has drained.

But dissipative handles on all metal tools that don't need handles like tweezers is expensive, posh, makes them hard to clean and gives them a soft component to wear out. I don't like it.

I think miniature plier and lead cutter-type tools are the main ones I would be concerned about since they have plastic handles and often make direct contact with components, but if their metal is touching the ESD mat before you pick it up then their charge gets drained. Same with screwdrivers and those rarely in direct contact with components.



EDIT:

I'm going to try to explain how ESD works in general instead of answering "what if?" one-off questions for every possible scenario. 

Something with a neutral charge will be defined as when something has a charge that causes it to have a voltage equal to ground.  As an analog, charge, is the amount of pressurized gas in a tank.

Then you have voltage of an object (always measured relative to another object), which is basically a measure of how hard the charge is trying to push through barriers in order to equalize itself. As an analog, this is pressure of the gas in a tank.

Charge equalization
When two objects touch, the charge try and equalize. This does not mean the charge will split evenly between the two. It means the charge will divide itself up between the two objects until the voltage between the two objects is equal. Just like how connecting two gas tanks together doesn't mean that they will share the gas evenly, unless the tanks are equal in size; the gas will divide itself up between the two tanks until the pressures are equal.

Objects themselves can obtain a charge (and therefore an associated voltage which pushes the charge to try and equalize it) that is positive or negative with respect to ground or with respect to each other.

Or in electrical terms, just large and small capacitors and ground is an infinite capacitor which follows all the same principles as all the mumbo jumbo above. Every object is a capacitor and every object can hold a charge.

Ground will be taken as an infinite charge sink. For analogy, it is taken to be an infinite gas tank. That means that if you dump a bunch of charge into ground, it gets spread around so much that its voltage will not change. Similar to connecting a finite gas tank to an infinitely large gas tank; In the process of pressure equalization, the finite tank might change a lot or a little in pressure but the infinite gas tank will change very little (i.e. zero) since the amount of gas being moved to equalize gets spread out so much within the infinite gas tank that the pressure does not change. This is a bit pedantic since we have defined ground as our neutral, our reference, so it is always zero no matter what happens. But it's not as pedantic when talking about other objects like a large metal shelf, or small metal tweezers. 

Materials are broadly divided into three categories. They all kind of look like this with R being different for different materials and with each capacitor being a localized area on the material that can hold charge:



Conductive- These objects have low resistance which means charges are free to move around within a conductive material. That means that any charge at one end is free to flow through the material and out the other end. This means that they can transmit an ESD charges from one end and damage something else on the other end. But it also means that it is difficult for them to actually generate a charge via rubbing since any charge built up into them can easily escape. Not to say that you can't impose a charge into a conductor by isolating it and then touching something charged up against it which will equalize/split the charge between the two.

Since charges can move freely within a conductor a larger conductor acts like a larger capacitance relative to a smaller conductor for the purposes of the definition of "equalize" mentioned above.

Conductors are like gas tanks with an open valve.

Dissipative
These objects have an intermediate resistance that still does allow charge to flow in a timely manner, but their resistance is high enough to limit the amount of current to a "safe" level. The result is that when an discharge does happen, enough voltage is dropped across the material and the current is reduced to a level that is less hazardous to the component.

And of course, since charges can move within dissipative materials in a timely manner it means that when in contact with ground (or anything else conductive or dissipative) they will equalize their charge over time, though not as quickly as a conductor will.

Dissipative materials are like a bunch of interconnected gas tanks with leaky valves.

Insulative
These materials have high resistance and charges cannot move freely through the material. That is not to say charges cannot move at all. They are the most complicated material out of the three. Charges indeed cannot move freely through an insulator, but they can move and everything becomes a conductor once the voltage is high enough.

If insulators were gas tanks, they would be like interconnected gas tanks with very slightly leaky pressure release valves that allow air to equalize very slowly but can also suddenly blow when the pressure gets too high. Since charge cannot move freely through an insulator you cannot think of an insulator as a single pressure-relief gas tank in the same way a conductor can be thought of as a single gas tank. You have to think of an insulator as a network of many very slightly leaky, small gas tanks with pressure release valves that can transfer air between each other (but with difficulty).

That is to say, that an insulator can pick up a charge and hold it locally somewhere on its surface. If it picks up too much charge locally, then the voltage gets high enough to push the stored charge around and spread it across the insulator's surface, which reduces the voltage with itself. But this takes time (similar to how charge can move through dissipative materials but more slowly than conductors), and there is also a limit since if the charge is spread out as much as it can be, and more charge is built up then the voltage just increases again. So if you build up too much charge or build up charge too quickly the voltage somewhere on the insulator will increase.

Increased voltage means the charge is pushing harder to equalize within itself, and more of our interest, into something else. So if a lot of charge is built up rapidly in a local spot in an insulator, the more readily that spot will discharge into something else. Given enough time, the charge will spread in an attempt to reduce the voltage will cause it to not so readily shock other things, but cause it to be able to shock things from areas other than where the charge was initially accumulated but with decreased readiness. It's basically like diffusion.







simulate this circuit &ndash; Schematic created using CircuitLab

So hopefully this just gives you a general background to qualitatively answer all your "what if" questions. You should be able to link these together in different scenarios.

Some examples demonstrating connotations you may not have thought about:


If an component has slowly built up a charge into it will it take
damage? (The answer is no if it was built up slowly enough). See
above how dissipative materials work to protect things.
What if I then place the component in #1 onto a conductive, grounded
surface, (or just a large conductive ungrounded surface for
    that matter), will damage occur? The answer is yes. Slowly building
    up charge into the component won't have damaged it, but the rapid
    discharge will.


Your specific question:

"Won’t touching the
    plastic handle or any material on the screwdriver ground it?" The
    answer is yes, no, and kind of due to the locality of charge in an
    insulator. 

Touching it to any material will kind of equalize the charge but
    only at the spot you touched it at. And it will take time (potentially a lot of time) if this other "any" material is not a
    conductor. Even if it was a conductor it would still take a lot of
    time for charge to equalize elsewhere in the plastic handle and
    reach the point of contact to leave the plastic handle through the
    conductor. Furthermore, note that that the conductor here is not
    necessarily grounded so you are actually charging up the conductor
    when equalizing charge. Both objects could still shock other
    objects, but with more difficulty and with reduced severity since
    the charge has been spread around a bit so the voltage is reduced.
    Then note that this conductor could also be the blade of the screw
    driver itself.

Other things you may not have thought about:


Such as, "if I have a screw driver whose blade already has a neutral charge because I grounded it, what would happen to the metal blade if I then grabbed the insulative plastic handle when my body was carrying a moderate charge? If my body was carrying a high charge? If I rubbed the plastic handle so hard as to continuously accumulate charge onto it?
What if I had a big metal ungrounded shelf that was at ground potential (i.e. I connected it to ground momentarily but then disconnected it) and touched a metal tool that was carrying a charge? Would the shelf become charged? By how much? What would the tool's charge be afterwards? By how much? How easily could either the shelf or the tool shock something afterwards?
What if I then touched a metal tool with a neutral charge to the big
metal bench in #5? Would it become charged? By how much? And how
badly would it be able to shock a component at that point