The only routers guaranteed to work with Time Machine in all respects (including the ability to restore a complete system) are the Apple AirPort Extreme Base Stations and Time Capsules. All of them have been discontinued.
Routers from other manufacturers may or may not work, and their claims for Time Machine compatibility have been demonstrated to be dubious at best, repeatedly, since Time Machine's inception well over a decade ago. The practical effect is that if they fail to work with TM in any respect, their manufacturers blame Apple's Time Machine software. Apple in turn will blame their manufacturers for failing to comply with Apple's Time Machine specifications. The only person who cares about your data is you so choose accordingly.
If your needs are such that a backup is required to be completely reliable, and you do not have or cannot obtain one of the above genuine Apple routers, then use directly-connected hard disk drives. I use a variety of Time Capsules / AEBSs and USB hard disk drives connected in every conceivable supported configuration. All of them have been 100% reliable in all respects.