Sounds like the display connector got damaged for sure, but if the digitizer still works, you can use a lightning to HDMI adapter to pull the display on a monitor, and just use that to navigate around the phone to do a backup.
Probably a failing board, you most likely need to just replace the board. For the price, its typically more worth it to just get a new device unfortunately.
I’ve seen this a couple times on an iPhone X, and it seems to be a board defect. When you say you tested the board connectors, did you test voltage on the pins specifically?
This is a known issue with iPhone 7’s. It is a board issue, nothing to be done to fix. Apple has a free replacement program right now. https://www.apple.com/support/iphone-7-no-service/
Sounds like either a dying battery, or just a simple boot loop (caused by corrupted software). I would try to restore the phone first, by putting the phone in DFU mode. If it doesn’t restore, its probably a battery issue or a logic board failure. Overall, its most likely a simple boot loop, and hopefully a restore will resolve it. Good Luck!
It is most likely a logic board failure. 99% of the time, an issue where you cant restore after trying 5+ times, it is a logic board failure such a short in the board, etc. This probably isn’t worth the time / money to repair the device.