I'm not clear about the difference between reboot and reset of Time Capsule
Reboot means pull the power out.. count to 10.. plug the power back in again.
Reset means push in the reset button with a paperclip etc.. for about 10sec until it flashes rapidly.
And why not rename TC the same name it has now?
Your naming of the TC is part of the problem. It is using old Mac protocols.. not new SMB protocols (i.e. windows). Apple changed the Mac to use windows networking. Only partially up to Catalina.. but much more seriously now.
So reset the TC and give it a new name.. e.g. TC is perfectly fine.
The rules are short.. <8 characters if possible, no spaces and pure alphanumeric. Do NOT accept apple naming which the setup wizard in airport utility will try and use.
Use password that is also short 8-20 characters and pure alphanumeric.
As noted.. no files on the hard disk are lost.
I think 90% of your problem is networking.
Messages -69842 and -69845 in case anyone knows what the codes mean.
Apple never publishes lists of error codes and what they mean.. but it strikes me that the TC has simply dropped off the network.. don't over worry about it.
Do the reset. Change the name as indicated.
If you still cannot get migration assistant to work there is another method.
You will need a USB drive to plug into the Mini of size equal to current boot drive.. or larger.
Boot to recovery and use Time Machine Setup Assistant. This will work better than migration assistant.
Recover the entire boot drive to the USB.
I gave extensive if somewhat outdated instructions here.
https://discussions.apple.com/thread/7236090
The first part is using Migration and the second is using Recovery.
Please NOTE.. for anyone reading or to remind yourself.. NEVER EVER TRUST TIME MACHINE TO RECOVER YOUR COMPUTER.
Before you wipe a computer create a bootable clone with a tool like SuperDuper or Carbon Copy Cloner.
And test the clone is fully operational by booting from it.. before you wipe the drive and do the reinstall. That ensures no matter what happens you can always get back to the working setup you had.