Here's a better choice -- same brand (Panasonic) of 1.5F caps that cost less.
https://www.greatplainselectronics.com/proddetail.asp?prod=CERS-1.5F-5V

The three pack of orangeness is your replacement battery pack. From CPU board photo at the 10000 ft altitude, it doesn't look like any damage from battery leakage but a closeup around the orange batteries may show differently. A previous owner may have saved you a ton of $$$ by changing out the old battery pack. *IF* they're NiCads and are installed properly then don't pull them yet, they may not be the issue. If diode CR34 is OK then, yes - maybe time to pull batteries. Easy way to check what batteries are doing. With power turned *off*, using volt meter - black lead to ground side of capacitor C1, use red lead to measure the voltage at banded end of CR34? This will tell us if there is any charge on the batteries and whether or not they're in backwards.
C1 is gray cap near the A1-J1 connection.
If you get about 3V there then measure voltage at the 5101 located at Z5 (where battery backed up data is stored).
Black lead still on negative end of C1 and red lead to pin 22. Ignore that blue arrow - that's pin 11 which is a data line.
This will at least tell us if there is a charge getting to the batteries and if the voltage is making it to Z5.

Now to see if the voltage is getting anywhere that it shouldn't. With machine still off - measure voltage directly across capacitor C1.

If you give up on this one - email me the back board from that machine! Mine is broken in the bottom right corner with a wimpy metal bracket holding it together.

Free play can be added to that machine only by changing PROMS at U2 and U3 but that isn't an easy or cheap task.

I can see there's a few other issues that should also be addressed but let's tackle this one first before moving to other things.

5101_Corners (resized).jpg

Howdy Chris - lift and swing out backboard. With mine, somebody swung the back board out then lifted, not pretty. But I was actually just joking about asking for his, I noticed how nice his looked compared to my band aided board. I really don't need one - as ugly as mine is, it works fine.

For your list of stuff, other than the batteries, you hit on my entire list of "a few other issues" that I was going to bring up when he got to that point.
I have only seen the black Data Sentry batteries, I thought that was all they were made with. My Ice Fever's CPU has had the batteries changed, very clean CPU fortunately but still has nicads. The ones I have... are orange, but they look much different from what he has. I keep close watch on mine since I have it open often for testing/verifying 'stuff'.

finnflash - for testing, you can disconnect the 7 pin plug on the small reset board down in the corner.
Going to NVRAM - yeah, probably easier than futzing around with trying to figure out what is going on with batteries. And safer in the long run.
Read up on the System 80 stuff at pinwiki as Chris mentioned, grounds weren't performed very well on these boards. As far as the power supply - I tend to go with the "if it ain't broke" idea here as these can be a bugger to work on at times due to the sandwiched heatsink.
And soon, you'll be cursing the Gottlieb edge connectors with the rest of us.