How can I add a ground wire to my older Square D panel?
No need to add ground wires. There are more serious problems. 

In  EMT/IMC/Rigid conduit, the conduit is the ground

This looks exactly like every one of my panels.  No grounds in sight and no ground bar.   And I get the occasional sophomore who goes "How old is this service?" "1960." "Oh noes, there are no ground wires!" "So?" "You need to put ground wires in all these conduits!" "No, you don't." 

Presuming the 1978 install date, plastic conduit was not popular at that time.  We know the branch circuits are conduit because you said you plan to add a ground wire.  We know the feeder is conduit because there are other circuits in there.  So I'm gonna bet that most if not all of these are metal conduit, but double check that yourself. 

I would speculate you're in Chicago, commercial (e.g. condo or converted space) or somewhere conduit is mandatory. 

If it's PVC conduit, then yeah, you need a ground wire.  

On to the problems. 

1. Grounds on the neutral bar.

I'm guessing this was done by a noob or Romex monkey unfamiliar with wiring methods here, and out of inexperience hucked the green wires onto "the bar, since 99% of what I do is into main panels therefore neutral==ground AFIAC".  

Which is wrong, as you observe.  You will need to ground the green wires.  One option is to drill and tap a  #10-32 hole in the service panel back wall, and ground them to that with a pigtail.  Another option is to follow the conduit down to the next junction box.  If the conduit and junction box are all-metal, then one of the holes in the junction box will be tapped #10-32. Stick a little green ground screw in that hole, pigtail off that, and ground there.  That's SOP for me when I make an EMT-Romex transition.  As a result my service panels have no ground wires at all. 

Another option is to get a ground bar for that panel.  

That noobness is a yellow-flag for other aspects of that work, so I would follow those green wires and carefully inspect all that for compliance with local Codes.  

2. Way too many wires in that conduit

There's a rule, 310.15(B)(3)(A), which applies to conduit over 24" long.  It  forces you to derate wires when too many are in a conduit.  For 120/240V split-phase, things factor out to this: 


2 or 3 circuits per conduit: No real effects.
4 circuits per conduit:  15-30A circuits unaffected. 2/0 copper limited to 150A. 
5-10 circuits per conduit: 15-30A circuits must upsize by -2 numerical sizes. 2/0 copper limited to 100A. 


Whatever you've got to do to get down to 4 circuits per conduit, go ahead and do that. Stick with my table above; if you deep-dive into the gory details of 310.15(B)(3)(A) remember neutrals are free in 240V split-phase circuits, and grounds are always free, so every circuit counts as 2 wires.  But "no more than 4" is the rule of the hour. 

3. Serving neutrals out of a different panel than hots

Oh heck, no.  

Your plan of moving the breakers so the hots and neutral come out of the same panel is just right. 

4. Seeing if the N-G bond is pulled

Pretty easy.  Throw as many loads as you can onto circuits in this subpanel.  Then use a DVM to measure voltage between the neutral bar and panel case.  If it's not bonded, you will see a small voltage differential, fraction of a volt.  

Now, turn the loads off (right at the breakers is fine).  The voltage differential should get significantly smaller tending to zero.  

If it was improperly bonded, the voltage difference would stay very close to zero and not move when you cut loads. 

5. All those missing grounds

A similar test can be done on branch circuits to see if they are grounded.  Measure the voltage between N and G at a socket. If the circuit is quiescent, you will see the same difference between N and G that's present at the panel.  Then, plug in a big load like a heater and watch it change. You will see the N-G voltage difference increase just a tick. 

I would expect them all to pass; metal conduit is pretty good stuff. 

6. 225A subpanel on 2/0 copper wire

130 mph tires on my 90 mph car.  Problem?  No. 130 mph is a "never exceed" rating not a mandate. 

This subpanel has no main breaker, so "225A" is the bus rating.  You are not allowed to feed this panel from a 300A breaker.  I rather doubt you are.  

What matters is the breaker on the other end of those 2/0 wires, which is what protects the wires and the panel.  I rather suspect it's 100-175A depending on how correctly they followed the conduit-derate rules.  I'm not expecting to find a problem here. 

7. Multi-wire branch circuits.  Without handle ties

I see 17 hots landing on breakers, but only 9 neutrals.  There appear to be five cases of black-red pairs landing on individual breakers. Now, mind you, "hot" colors cannot be relied on to be meaningful, but I feel most likely these are multi-wire branch circuits (MWBC). 

These two hots seem to be individual circuits, but they share a neutral.  That makes them a single circuit.  

Modern MWBC rules absolutely require you to put these either on a handle-tied or 2-pole breaker.  So that when you shut off the circuit for maintenance, both hots shut off.  The two hots must be matched ones sharing the same neutral. They probably were grouped adjacent once, but as breakers get moved around, who knows?  

Also, the two MWBC hots must be 240V apart, i.e. on opposite poles.  If they measure 0V apart, that means current from each leg will add, and that will overload the neutral. 

8. Wires not identified as relating to each other

I see a glob of black, red and white wires heading off into big pipes.  With no evidence to tell you which hot-neutral or hot-hot-neutral group goes together in which circuits.  

That's illegal, for so many reasons. Just you wait, when you start putting GFCI breakers in here, you're going to need to know which neutral goes with which hot.  

So I'm afraid you're going to have to get a clamp meter and heater, and make a map, and figure out which neutral(s) return current for which hot(s).  Once they're identified, wrap the groups of wires with tape or shrink tubing.   Since the last guy only used black white red, you might as well get multi-colored tape and color each wire of your MWBCs: the blue group, the green group, the yellow group, the brown group, the orange group, etc. 

With THHN wire in conduit smaller than #6, the native color defines hot/neutral/ground: putting colored tape on them does not change their function. Blue tape on a white wire = neutral in blue group. 

9. Wires too short to reach every breaker in the panel

Oh wait, you don't have that problem. Don't create that problem.