Should we give up on trying to back up our M1 MacBooks to our Time Capsule?
Probably. There are numerous reports of M1 Macs having difficulty backing up to a Time Capsule.
Has the Time Capsule product been discontinued?
The Time Capsule was discontinued in April 2018.
If so what kind or router will we need to replace the Time Capsule?
The Time Capsule was a special product that combined a router and a hard drive in one single package. No other manufacturer ever offered a similar product, so a direct replacement is not available.
My suggestion would be to look for a new router and then connect a USB hard drive directly to each Mac that you want to back up.
If you want to back up your Macs over Wi-Fi, then you might want to look at the Synology router, to which you can connect a USB hard drive for Time Machine backups. While this would be two separate products that work together, it is the closest that you can come to a Time Capsule which was really two separate products combined into one package.