Restoring from Time Machine on Mac without video card
You are almost there. You could treat either the whole machine or just the system drive as an external HDD and attach it to your MacBook.

Attach your Mac Pro in TDM (target disk mode) to your MacBook. Boot into the Mac Pro’s system from your MacBook. Repair what you need to do.


Connect both computers.
Shut them down.
Power on the target Mac in TDM by holding the T key.
Power on host Mac with Alt/Option key pressed until you see MacOS boot manager.
Choose the target’s system disk
Boot
Repair graphic drivers, etc.