How to troubleshoot 2013 Mac Pro without working display? (Recent MacBooks + iPads available if helpful; Catalina/Big Sur)
Did you have Back To My Mac or Screen Sharing (VNC), Remote Login (ssh), or File Sharing (AFP/SMB) enabled on the Mac Pro beforehand? If so, try to connect to it over the network from another device. That’s the only way I know of to use a second device for I/O.
Does the Caps Lock key light up when you press it? If so, this is a good indicator that the Mac is responding to events in either the OS or at least in firmware.
Can you try to log in to the Mac by “muscle memory” (i.e. use your keyboard to enter your credentials like you used to) and use the Volume Up / Down keys (possibly with the Shift key) to listen for the UI feedback sound. This also would show a good OS.
Try going into Boot Picker (hold Option during boot) to test video out from firmware, in case this is an OS-specific issue.

Unfortunately you’ll probably end up having to buy something in order to isolate the failure: Maybe it’s the display, maybe it’s the cable, or maybe it’s the Mac. (My money would be on the former two more likely flaking out than the latter.) Might be a good excuse to invest in a USB-C adapter for your newer devices.