How can I test the reset circuitry? The issue seems to be when boot from cold, but once booted its mostly ok (maybe the odd recoverable alert though).
Weed
This is not straightforward on an a3000T. Ideally means removing the motherboard. And tracing the signal with a logic probe. Working from a schematic.
On powerup, the motherboard reset signals waits for a shortwhile, then applies a momentary low pulse into the Rst line. This gets fed around the board to different components.
Chips with power on them start in an "undefined" state, Rst clears them to an initial zero point. The keyboard circuitry for reset feeds ino the same system.
I think you have more than one problem here, so probably a good idea to just reset on power up, then look for the memory fault.
The think with cleaning connectors is - you have to clean both of them, and oxidation builds up on BOTH sides.
Board cleaner does not remove oxidation from pin contacts and solder points. You have 30 years of copper "rust".
OK, this may not fix your memory problem, but it will help to keep a machine reliable.