Well, I've swapped the CIA's, now I have a gray screen, machine doesn't boot anymore.
That (or usually stuck on the white screen) can happen when one of the the hardware timers in U7 isn't working. But can be caused by various other things too.
I swapped the CIA's again to their original position and the gray screen keeps showing.
There's possibly some intermittent problem that has been introduced when flexing the main board. The most common intermittent issue is with IC socket contacts. Try removing then re-inserting all of the socketed dual in-line ICs which is often an easy fix. See if you have any change.
Failing that, the hardware should be working enough to run one of the debugging tools available, John Hurtell's DiagROM would be useful at that point. You'll need a copy of it in a physical ROM.
There's nothing definitive yet to say if you have a failed CIA or not.