Your neutral is weakly connected. Period. Stop.
This means that your DMM/ Analog meter -- which draws virtually no current -- gives you nice readings.
Then, upon any serious load ( Hot to Neutral ) your device is starved for electro-potential.
Cut the problem in half: turn MAIN off and re-verify that all load side connections are sweet.
Then you can begin going back up the line towards the Poco. The neutral can get fouled up at any point from pig to MAIN.
The weird C/Bs you removed were most likely smoked by the bad neutral.
Use a megger to check out circuit quality -- and verify that the circuits haven't been 'repaired' by a handyman.
Older panels must also be inspected/ tested for corroded buses. I've seen resistances as high as 3% of full voltage from rail to C/B terminal. (!)
You can be getting hell from both ends in an older panel; so methodically check everything out.
PS: Always check for crossed hots.
These are the fellows that produce tools for other name brands you know and love. ( Use your imagination. )