Time Machine fails to eject disk (sparesebundle) on Time Capsule, then cannot mount on different Mac
You may not like my advice, but walk away from network backups. Use direct attached storage and use two backup destinations if you can’t afford to have one take days to check the filesystem if they eject due to a cable failure, power failure or cat.
To have things run as you describe, you’d need dedicated new hardware, have to set up a directory service and/or managing permissions by user ID, manage SMB and possibly AFP sharing services, the source system has to do a ton of overhead to manage the storage bands. Time Capsule is not powerful enough for the task in my experience. It was fine 5 years ago (fine for light use, OK performance) but not anymore.
What’s easy is APFS filesystem on the Mac so your get snapshots and HFS+ filesystem on the storage destination.

And to answer your question, you can use tmutil stopbackup to ask for a quick abort, but since you skip any clean up, your next backup interval takes so much longer since the code has to do all the cleanup before it finishes the next interval. I would need to analyze the network capture, but temporary unavailable errors can be due to the server not responding in a timely manner since it’s busy managing files and network as opposed to serving the data to your Mac.