How to troubleshoot 2013 Mac Pro without working display? (Recent MacBooks + iPads available if helpful; Catalina/Big Sur) Briefly: Is there any way to test a Mac Pro (2013, cylinder model, Big Sur) without a working display — either without a display, or using a recent USB-C MacBook (particularly MBP 2019, MBP 2020, and MBA 2019, Catalina/Big Sur) or iPad (Pro, 2017 and 2020) as a display, either wirelessly or over USB?
Long version: I’m trying to troubleshoot a Mac Pro + HDMI/DisplayPort display setup, which has been working fine for several years, and stopped working last night.  Both Mac Pro and display individually seem to power on as normal, but the display just gives a “no signal detected” message.  (Precisely: when the Mac Pro’s power-on button is pressed, its front lights come on, its fan whirs up, and connected external drives also power up as usual.  Similarly, when the external display is powered on, it shows its manufacturer startup screen, as usual, but thereafter just a “no signal detected” message.)
We have several USB-C MacBooks in the house, but no suitable adaptor from HDMI or DisplayPort to USB-C, so we can’t test the monitor with them for now.  So I would like to check that the Mac Pro itself, to see if the problem is on its side.
Is there any way to test/troubleshoot the Mac Pro, either with no display, or using one of the MacBooks or iPads as a display or terminal?  The MacBooks are all recent models (≥2019), and on either Catalina (10.15) or Big Sur (10.16).  The Mac Pro is from c.2016 (i.e. the 2013-onward cylinder model), and running Big Sur.  They are all normally on the same wi-fi network, and one of the iPads is set up with the same Apple ID as the Mac Pro.  We also have USB cables that could go between the Mac Pro and any of these devices.  The Mac Pro is fully backed up on external hard drives.
I have found earlier related questions suggesting Target Display Mode for this, but that seems to have been discontinued since High Sierra (10.13).
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.