The very fact that Airport Utility is still in the OS means you can still use the Time Capsule..
BIG FATT UGLY BUTT.
Catalina is not behaving very well network wise with the TC.
You may not have had issues but let me assure you plenty of people have had big problems since 10.15.0 and every patch thereafter. (As well as going back several versions of OSX. Catalina did not start the rot.. it just completed the structural members failure rate).
If you want the TC for Time Machine.. well the day may have arrived when that is not such a great idea.
You need a NAS for network backup that supports SMB2 /3 since SMB1 and AFP are only protocols Apple have ever supported on the TC. And Apple is trying their darnest to bury AFP. SMB1 is a huge security hole.
Synology NAS are proving quite good. Of course Apple can ruin that anytime they please.
How old is your TC? If it is 4 years I would just recommend replacement. It might go another year.. but Apple is happy with 3 years. That ends the issue.
I would say you can still use the TC under ideal network conditions with a better backup software like Carbon Copy Cloner. Better in that it is based on excellent rsync commands with a decent GUI front end. And is much more tolerant of non-ideal setup.
First step would be to get the TC working IDEALLY.
Reset the TC to factory..
Open the Airport Utility in your new Catalina and it should still discover a reset TC.
Go through the setup..
However use short names.. <8 characters, no spaces, pure alphanumerics.
Passwords the same rules but 8-20 characters.
I do not know if your TC is main router or bridged.
If bridged to 3rd party router it is best to set a static IP on it. So it can never get lost again.
Turn IPv6 to link-local only.
And fix up the domain if possible. It may not be but TC is local whether you like it or not.. so if you still use the TC set the main router to local as well.. it is minor point but is another factor that helps.
You can also then mount the TC hard disk by IP address instead of TC name.. this is another help when main router is not so bonjour aware as it could be.
Can give details on any of the above if you need them.