If “the solder joints are always bad” is rule #1 when repairing old gear, “the power switch is worn out and must be replaced” is rule #2.
The TI-99/4A in-console power supply has a DPDT switch that handles +5VDC and +12VDC. The switch internals will wear grooves in the contacts over time. After nearly forty years, that power switch is going to be grumpy as hell. It will not reliably make nor keep contact when energized, and should be replaced. Desolder the sucker, take it down to your local parts house, and look for something that is close enough to work.
The closest replacement I could find in Akihabara was a non-PCB-mount switch. The actuator is a bit taller than the original, and I had to bend the mounting lugs out ninety degrees to make contact with the PCB pads, but the console in question now powers on reliably. The only thing missing is the tactile “thunk” when switching it between “on” and “off” positions, but that’s an aesthetic thing that can be ignored.
The TI-99/4A console power switch is like a VW shift hanger — if you haven’t replaced it, or even looked at it, then it is bad and will cause problems. Just replace the (redacted) thing as soon as you open up the console, and you’ll save yourself a lot of heartache down the road.