Routers compatible with Time Machine 2021

A better way to think is a NAS (Network Attached Storage). The router is then irrelevant and the NAS takes the task of backup. This has several large advantages.

1. A NAS is designed strictly to do one task well. Store files and serve them to the network.. with speed and reliability. Hence it can back itself up, it can create redundancy with multiple disks as well as offer full speed. A router is designed to route ethernet and wifi between a local network and internet as primary task. It is only secondary add on to do backups to USB drive.
2. NAS has memory and processing power to use SMB3 which is now Apple's preferred protocol. Routers do NOT. A router that supports Time Machine uses AFP protocol which is still available but is not best way to handle Time Machine since High Sierra. SMB3 is very demanding and routers do not have the resources to support it. The vast majority of routers have SMB1 which is now deprecated. That includes Apple Time Capsule btw.
3. NAS has better service life than vast majority of routers. They are expensive but should give many years of service.
4. NAS from Synology in particular is our preferred network target for Time Machine backups. Synology has gone to considerable lengths to make their products Apple and Time Machine compatible.

If the cost of the NAS is too high even a second hand recent 2 disk version from a couple of years ago will do nicely. You don't need to the latest greatest.

Synology also make a decent router RT2600AC which in my testing was the best by far with Time Machine. The problem is the model is expensive, several years old now and so far no AX replacement. No idea if Synology is going to continue router production.

High end Asus models also worked with this proviso. You need to be prepared to change to another backup software like Carbon Copy Cloner if the Time Machine backup proves unreliable. In testing the Asus would fail on about a 2 month cycle. It would corrupt the backup and need a wipe and restart. I tested this for over a year. CCC had no issues but I did not test as long. YMMV. Firmware is continually updated and I have seen few reports of people testing on long term basis. The problem with all the standard sites is they test for a few days and think that proves the matter. IMHO that is useless.. if it is not going to work long term.

A few other brands offer Time Machine.. Some models of Netgear routers do indeed offer it.. Not sure what one you bought but people have had issues. I noticed a fairly cheap TP-Link offered Time Machine recently .. But their firmware is seldom well optimised and disk speeds are poor. I would not recommend it.

Other options.

There are projects where people use a Raspberry Pi with a NAS firmware.. apparently works ok. There are other DIY NAS projects.. Again YMMV.

A Gen4 Time Capsule.. with an external power supply I reckon could go for a few more years.. I have fixed many of them.. and while you will get the occasional bad one that cannot be repaired, they do work really well as purely a backup plugged into your network. Sell for about $50 now on eBay.. whereas the Gen5 is around $100+

Some DIY skills needed but overall pretty easy. And still better IMHO than the later Gen5 which is internally built to self destruct the moment you open it. The Gen4 was solid.. with proper SATA cable and decent HDD mount points.. it will take 8TB disk no problems. If you want a cheapie solution for a few years.. kick the can down the road so to speak.. could be worth consideration.