That ground bar
The proper way is neutrals on the neutral bar, and grounds on the ground bar, and a neutral-ground bond. I'm a big fan of using a wire for that, because a) it makes burnout much more obvious (a screw head can hide a vaporized screw shaft), and b) you can put a clamp ammeter on it, which will reveal any ground leakage (ground faults). Very useful for troubleshooting.
Builders (i.e. people who build housing units by the thousands) growsed about needing 2 bars since they're connected to each other, so they got NFPA to "bless" an ugly hack where they use one bar for both. This has been the source of staggering misconception, and is the reason my catchphrase on this forum is "Neutral Is Not Ground". Because so many people believe neutral is ground, based on what they see with their own eyes.
Obviously your first exposure was to "ugly hack" type panels, and you've normalized that. No big; just de-normalize it, and know that logically, it is important they are separate.
Given that you have the opportunity, I would separate them completely, fit a hefty ground wire or two between them, and remove the ground bonding screw because as said, I don't trust them.
Imperfect, but not so bad
Since you are searching for bad technique in that box, I see only a few minor issues, mostly style matters.
- Ground screws double-tapped - this is not allowed per GE labeling.
- "Mr. Snippy" cut all your wires short, so you won't be able to move breakers around in the future. Best practice: Leave enough wire so both hot and neutral of every circuit can reach any breaker space in the panel. Neutral for GFCIs.
- way too small a ground bar. This panel is made to be double-stuffed, and can support 40 half-breakers. Best practice: ground holes > possible breakers.
- way too small a panel - this panel counts on using GE's unique "double-stuff" breakers. *Best practice: 40 spaces, since now-required AFCI or GFCI breakers take full spaces.*
The homeowner probably added the ground bar because he knew he'd run out of space lickety split in the neutral bar, or because he's a class act and likes to do things right. He added a small ground bar because that's what GE sold at the time. The Sentry/Sense meter had nothing to do with it; that came along much later as it didn't exist when that ground bar was fitted.
Better ground bars
They really shorted folks on those neutral screws. I wonder if this panel dates back to when grounds were rare (1960s). Nothing wrong with that (except the shortage of grounds obviously).
On the ground bars, I would not hesitate to fit multiple, larger ground bars. Most panel parts must be UL-classified for this panel or it will cause a real problem; however alien ground bars should be safe (if not quite code). See if you're lucky and a 3rd party ground bar will line up with the holes on the lower left (make a tracing). Otherwise just drill and use #10-32 self-tapping screws (10-32 mandatory as ThreePhaseEel discusses). I'd put the ground bar 1.5" from the edge of the panel.
I'm no fan of "screws as grounding path" so I would wire the N-G bond with nice hefty ground wire. I would not run 2 ground wires N-G1 and G1-G2 - if you want to do that, run 1 wire from N through G1 to G2 (hence setting it back from the edge). That way G2's path only involves 2 lugs instead of 4. Given my penchant for running ground wires, why not just use sheet-metal screws? Because some future installer might decide my wire is stupid, and remove it.
Notes on that panel style
GE is the last manufacturer still making a system with discrete half-breakers. (nothing wrong with the concept, it is coincidental that the other makers cheated their UL tests or otherwise made dangerous panels).
Each "stab"/blade has the usual horizontal blade normal breakers clip onto. This type also have a vertical "cruciform" made for GE's half-width breakers to clip onto. Note the "50" in top right is a 2-pole breaker grabbing the cruciform off the stab above it, and the cruciform off the stab below it. It is actually "riding between spaces".
This is a slick system, but it's very important you pay attention to "stab limits" since each stab can have 4 breakers on them. That 50 shares its lower stab with a 20. The leftside breakers are unfilled but imagine there were two 50's there -- 170A on a single stab - whoops! The panel labeling should discuss stab limits.