Welcome Jonas!
If you wanted to test just system powerup, and eliminate the keyboard being possibly broken from the equation, you can use the information in the link:
https://www.netbsd.org/ports/next68k/faq.html#no-monitor to just power up the cube from the DB19 port directly. I have done this on my slabs to confirm basic power-on without a keyboard before. You can also potentially just check the resistance across these pins off the cable that goes to the cube from the monitor and see if you see the change when pressing the keyboard's power button.
It is also important to note that the 3 volt BR-2/3A battery must be installed and operational for the system to power up. So you should pop your battery out and check its voltage. If the battery is bad, or missing, you can still source the BR-2/3A battery through a electrical component supplier or a seller that specializes in batteries. However, in a pinch, you can use a CR123A battery as a replacement. It is a bit too large so the fit isn't quite correct but at least would let you see if its the source of your power-on issues.
I am sure there are some other tips and tricks this community can provide to directly test the PSU, etc. But hopefully these two simple tests may help.
Good Luck!