Ive not serviced the PSU. its where I'll start looking.
This issue does have signs that it could be related to excessive noise on the 5V supply, or possibly heat related. If you have some way of looking at the AC component on the 5V supply (e.g. an oscilloscope), you'll immediately see the state of the power supply. Expect less than 50mV of AC noise on the 5V line when measured at the main board power connector, and the DC component between about 4.7V - 5.2V.
I've just noticed that you have a CPU board. To rule that out as a possible contributor, I'd suggest operating the A3000T with the onboard 68030 and memory and *without* the CPU board fitted to see if there's any change in operation.
What were you hoping to achieve with the reset line? Unlike most Amiga models, the A3000T reset signal is generated within the power supply, not on the main board. At power on, reset stays low until the 5V supply is stable, then reset is set to 5V and the system runs. Worth noting with the A3000T is if reset rises too quickly (it isn't delayed when the 5V supply comes up), then the fast memory controller (Ramsey) isn't initialised correctly, and you can end up with 0MB or 4MB of fast memory instead of whatever you have fitted.
If the main board has no sign of booting, then it's worth checking that reset/'power good' is at 5V for starters. But whatever is happening on the reset line won't be causing your software errors.
Also on the A3000T, reset is connected to the front panel key switch. I had a customer recently where the key switch had been butchered and was permanently holding reset low.