I need to back up the way Time Capsule used to, post-Catalina

So, Time Capsule and Time Machine together used to be the perfect backup solution for my household's Macs. Then they stopped working. After I got a new laptop with Catalina, I would regularly get an error message saying that the backup needed to be deleted and was starting over. Sort of defeats the purpose of a backup when it gets regularly deleted. Then my wife started getting the same error on her computer... and then it got even worse on mine, where it just wouldn't back up at all. Since my wife is also getting problems on her computer, it doesn't seem like the issue is with my setup. I started looking for alternative backup solutions. I tried Carbon Copy Cloner and Super Duper, and they both took an interminably long time... and then failed anyway. I would imagine that a big part of the problem, if not the whole problem, is just that Apple isn't supporting Time Capsule anymore. But it *is* supporting Time Machine, I think. So it _looks_ like [what I need is a good third-party wireless router](https://www.cultofmac.com/545927/use-time-machine-without-airport-time-capsule/) that will support Time Machine. But before I blow hundreds of dollars, I wanted to check in with people who know what they're doing. If I buy a good third-party Time Capsule equivalent, will that allow me to go back to the old way of using Time Machine? And are there particular things that I should look for in the hardware to make that happen?