I had problems with an STE which had been put away working, was dusted off after about 15 years, worked for about 20 minutes from switch-on and then died. Long story cut short, it was due to pins on the square CPU socket failing to make contact with some of the corresponding contacts on the CPU, although this was not visually obvious and the fault even survived removal, inspection, cleaning and reinsertion of the CPU.
The sockets chosen by Atari have a non-standard 'weird' pinout which is different to that of the majority of easily available PLCC sockets, so this is an added complication - I bought a genuine Atari socket from Best over in the USA, at great expense. Shortly after that, Exxos produced a useful little adaptor PCB which allows a 'normal' easily available PLCC socket to be used in Atari PCBs.
But anyway... assuming the availability of a scope and a needlepoint probe, a steady hand and a bucket load of patience, one way to diagnose or rule out socket problems is to start at pin one on the chip, look at the signal, then move to the pin 1 contact on the socket - both are accessable next to each other from the top side, but the probe point must be fine enough for you to be able to choose whether to monitor the pin or the socket contact, not both at the same time. Don't put any pressure on either the pin or the socket contact, use only the lightest touch needed to get a picture of the signal.
Essentially, if you see the same 'picture' on each pin and its corresponding socket contact then they are OK, but if you find any where the 'picture' on the pin and the 'picture' on the corresponding socket contact are different, that pin / socket contact is faulty.
The inherent problem with the PLCC sockets is that the four sides have a lot of pressure applied to the insides by the springy pins of the IC, but there is no equal and opposite pressure on the outsides of the four sides of the sockets so over time, the sides of the sockets are prone to bow slightly outwards and the connections in the middle of each side in particular may lose their connection.
Chandler, time you had a scope. Even an old 20Mhz or 40Mhz analogue CRT scope (Hitachi V212 / V412 or Hameg HM-203, etc), would be better than not having one at all.