Grounding Upgrades While Replacing/Adding Panels
TLDR:

No need to upgrade main panel. A big subpanel will support your future needs.
You MUST upgrade your service (and service entrance) to 200A since your main breaker is already 200A.  WTH?

This will handle everything you plan, and an EVSE also.


LOVE your EMT and rely on it for grounding. Consider it for your next run!
For #4 copper wire running in EMT conduit, upsize the 70A breaker to 90A. (instant capacity!)
That other bar is a neutral bar. Grounds are OK there in a main panel.
Having reclaimed your other neutral bar, remove the double-taps in the main panel.
Never again use a grounding rod as a substitute for a ground wire!

On the main panel
I see the installer made several questionable choices even if they are not Code violations.
Oh wait, there are code violations at the double-tapping of some neutrals, as well as the shared neutral lug.  The bare neutral to the right appears to be your service feeder neutral.  The insulated seems to be your subpanel neutral, and it belongs on the neutral lug on the left!  (but don't runt off any more length please).
Also, I see a 200A main breaker.  Given your modest loads, since you already have 200A, I wonder why on earth you need a service upgrad--- Oh, wait.  Those service wires look awfully small to be 3/0 Cu.   It looks to me like you have a 100A or 125A service with a 200A main breaker.   Yeah, I would go ahead and upgrade that service to 200A A frikin S A P before you set those service wires on fire. But the good news is, you don't need to change this panel!  It's not as full as you think, and your &quot;subpanel&quot; strategy has a lot of legs - you'll be fine!  To repeat:
The installer chose to clip off all excess length, in the goal of &quot;neatness&quot; (and nobody gives out awards for that).  This creates a great inconvenience (wires are too short to freely move breakers around) and also creates a thermal problem with so many active conductors bundled together, see neutral wires bound together.  Never do either one!! Every hot and neutral should (not a Code requirement) to reach any breaker space in the panel (neutral for AFCI or GFCI).  Unbound and slightly loose wires creates the airspace between wires needed for cooling.
You have #4 Cu wire coming off a 70A breaker.  Now, if you look in the ampacity tables, at 60C thermal, #4 is allowed 70A, clear enough.  However, wires run in conduit are permitted 75C thermal.  That gives your #4 feeder 85A, and you are allowed to &quot;round up&quot; to a 90A breaker.  So we change that feeder breaker to 90A immediately, and just bought ourselves 20A of headroom. Easy peasy!
You can squeeze 4 more full spaces out of this panel by doing more double-stuffing.  That'll give you room for full-size breakers for 2 more subpanels.
Remember if you're staying with #4 Cu, that breakers at 90A if run in conduit.   You should also evaluate aluminum for long runs, there's nothing wrong with it for feeder.
Neutral and ground bars
GE has a funny way of doing double-stuff breakers.  They sell 1/2&quot; width breakers that clip onto a &quot;cruciform&quot; bus stab unique to GE.  So this is a GE Q-line panel for sure, and given the positions, it's clear all the spaces have the cruciforms.  This panel has 16 spaces and can accommodate 32 circuits via the 1/2&quot; double-stuffs.
Since there's no such thing an accessory neutral bar, it would be a grave oversight for a panel manufacturer to short you.  Indeed, the bar on the left is a neutral bar that the installer has chosen to populate exclusively with grounds. It is legal in a main panel to put grounds on the neutral bar, but this has led everyone into an error in thinking.  Simply evict the grounds from your other neutral bar and you'll have plenty.
Here is the spotter's guide: The steel case is grounded. Neutral bars are designed to be possible to isolate from the steel case. Ground bars cannot be isolated; they are bolted direct to the steel case.  That's what you do; bolt on accessory ground bars anywhere you please. Either

a) use #10-32 (or any -32 or finer thread pitch at least #8) self-tapping screws to mount them, or
b)  run a fat wire from the ground bar to the neutral bar, ideally in a place you can get a clamp ammeter around it.

Ground rods and grounding
Your plan to update ground rods is a good one. Remember there is such a thing as an UFER ground - this is poured into the concrete and brought out as a chunk of reinforcing rod etc.  This is the best ground available.  Look for this on each concrete pour.  Other than that, if you are not confident of your ground bars, by all means put modern ones in!  They must feed into the main panel to count.  You can keep the substandard grounds connected.

At the back of the house, about 12 feet from where I'm putting in the new subpanel, there's a 1/2&quot; ground rod which I drove myself a few years back when I found that an outlet where I wanted to place a computer did not have a functioning ground

Never do that! Ground bars only handle lightning and ESD (which was what you were after in that application). They do nothing for ground-faults of human-generated electricity, because dirt is too poor a conductor of electricity to return 30A of current.  So you created the illusion of ground-fault safety, but none existed: you could've corrected that by GFCI-protecting the outlet, gaining its ground-fault protection along with the rod's lightning/ESD suppression.

although the EMT provides very excellent ground continuity

Fixed that for ya :)  Seriously, I once accidentally returned several hundred amps of current through 1/2&quot; EMT which was 50 years old and deteriorating from water damage.  EMT is a fantastic grounding path.  And you want a fantastic ground, because the better the ground is, the more likely a bolted fault will magnetic-trip (insta-trip) the breaker in 5ms instead of waiting 500ms for a thermal trip.
I know you want to use PVC for your next run because it's the &quot;new shiny&quot;, but that's because it's cheap (and not in the good way).  In commercial installations EMT is mandatory and PVC is not allowed. Consider staying on-board with EMT.  It is superior unless you are going underground. And yeah, it handles the ground for you.  I manage 9 buildings with over 2 miles of EMT - not a ground wire in any building. Our 800A service is grounded through two 4&quot; EMT pipes.  This, plus stranded wire, makes life much easier!
On speccing subpanels
We cannot say it enough: Go Big.  Go really big.  Going with a smaller panel saves you -- maybe, a latté.  Will you still be able to taste that latté several years later when you are confronting another &quot;box full&quot; situation?
Figure on this new subpanel (or a fourth additional subpanel) taking heat/circuits off this existing semi-full main panel.  There's no problem with your 200A service supporting four 85A subpanels like that, since the chances are bascially nil that they'll all be maxed at the same time.  85A = #4Cu in conduit.
Given your situation with your panel (plennnnty of amps; precious little space) I would use subpanels as your primary expansion method.  I like to see houses finish a subpanel add with about 50 spaces (remembering those Q-line double-stuff breakers take half a space). This panel has 16, your two other subpanels have ??? and ???.
Feel free to stay GE, it's a perfectly fine panel. But don't count on the Q-line &quot;double-stuffing&quot; method; most breakers these days need to be  AFCI, GFCI or both.  AFCI must happen in the panel not the first recep (unless the run to that recep is EMT conduit, haha, I love that stuff!!)  NEC 2020 brings GFCI requirement for 240V circuit and that can only happen in the panel.  Together these rules are basically the death-knell of double-stuff breakers.