Which routers are known to work with iMac Time Machine?

You ought to be able to connect the Time Capsule's Ethernet port to any router's Ethernet network, and join that network in client mode.

To reconfigure the Time Capsule to use that connection in lieu of its existing Wi-Fi connection, you will need to connect its Ethernet "circle of dots" port to the router first (either directly or through a switch) followed by a "hard reset" of TC, use AirPort Utility to configure it, and finally "reconnect" each Mac's Time Machine backups to that Time Capsule. Leave any USB hard disk drives connected to the Time Capsule's USB port, but you will need to "reconnect" to them as well.

If you don't "hard reset" the TC it will insist upon using its existing wireless radio. I haven't found a way to convince them otherwise. It will also want to "extend" that Ethernet network wirelessly, which you might want to do, but you can turn that off if you wish.

Those actions will have no effect on the Time Capsule's existing backup history.

References:

• How to reset your AirPort base station - Apple Support: refer to Hard reset.
• AirPort Utility User Guide for Mac - Apple Support
• The "reconnect" procedure (my terminology, not Apple's): Monterey - Time Machine not backing up… - Apple Community. That link will go directly to the relevant post.

Be advised I only use Apple equipment for all my networks, so I can't guarantee it will work with non-Apple routers, but I can find no reason it won't work with any other equipment. A Time Capsule's hard disks are networked drives like any other, but the resident NAS software on the TC is exclusively Apple's.

Besides the Time capsule, your options are a Time Machine compatible NAS or an external drive connected directly each Mac.

see > Types of disks you can use with Time Machine on Mac - Apple Support

some synology nas are suppose to still support timemachine not sure if that goes for their routes too

but if it was me then I would just connect an external thunderbolt or usb3 SSD drive and use time machine that way, it's sustainable transfer rate would leave anything over the network in the dust.

Why have a network if you can't access a remote storage device plugged into the router?

Good question. I seriously doubt that Netgear tech understood your question or knew what he was talking about.

As its name implies a NAS device is a network device like any other... a Time Capsule, a Mac, or any computer for that matter. And it's about as complex a device, meaning it will be running its own operating system, etc.

But this is straying off topic quite a bit. You can certainly use a non-Apple NAS device for backups, but if you want to do that I suggest using something other than Time Machine.

90 minutes... that's nuts.

Besides, I still don't believe them. What other NAS won't they support?

What apps are you using that require more than Gigabit Ethernet?