Setting up a NAS for Time Machine Back Up - Big Sur

I assume that there is nothing special I need to do in Time Machine to see network attached drives?

You need to ensure your NAS supports the "Time Machine over SMB" specification (usually that means it is using Samba 4.8 or later). Your title states "NAS," but you keep saying Router in your responses. There are almost no routers that support Time Machine over SMB. Most don't even support anything but SMB 1 which is completely insecure. Since it is internal to your LAN, the security piece doesn't really matter, much. However, any SMB 1 implementation would not support the Time Machine over SMB specification. If your router supports SMB 3, then it might have support for the Time Machine over SMB specification.

The file system of the drive does not matter short of needing to support creating a Sparse Bundle Disk Image on the drive. Time Machine will create a disk image on the network share which uses the correct file system it needs. The source and destination file system format do not need to be the same. You can back up APFS to HFS+. The file system is not transported, only the file data.

I'm sure the router has no idea how to handle APFS. A big clue is it thinks it is FAT16. You need to format the drive per the NAS specifications.

Generally not. USB ports on routers are typically used for updating the router firmware or sharing a USB printer.

If you want other machines to "see" the hard drive, and they are Macs, you can add the drive to file sharing in the Get Info of the drive in the Finder, and enable file sharing. Mind you the speed of the file sharing is depending on the speed of the network itself. WiFi in general only sees speeds up to 300 Mbps. If your router is hooked up via ethernet to both machines, you'll see up to 1000 Mbps usually, which is about 125 MBps, and that's if the route is not directing any traffic to the internet or other computers at the time. For every simultaneous data transfer, that cuts down on the bandwidth.

SMB file sharing is also available for Windows support on a Mac network. I'm not entirely familiar with how that will work, but it is an option that may work for you.

Time machine though has to be on a Mac formatted drive.

FTP is also an available transfer medium over the network for individual files that is platform independent.

While many routers allow you to share out a drive, it is almost always via SMB1 and that won’t have support for Time Machine.

Your router documentation should tell you what networking protocol it uses.

If it actually supports SMB3, then you might have a hope of making that work.

FAT32 does not carry over resource forks common with Mac files.

If you need the NAS to share data with Windows PCs, that's one thing, but if it is only going to be with a Mac, you may be able to format it HFS Extended Journalled.

If not, I would contact the NAS manufacturer for some Mac specific tools.

The most commonly used NAS storage device on the Mac, is one Apple used to make, known as a Time Capsule.