Time machine router function

Do you mean Time Capsule? Time Machine is just backup software. A Time Capsule is hardware composed of a disk drive to use for Time Machine backups and a router for both Wi-Fi and Ethernet. If you already have a router then this could replace it.

Does the Time Machine function as a router via the ethernet inputs

Time Machine is software on your Mac that is used to back up the contents of the Mac to another hard drive.

You might be asking about a Time Capsule, which is hardware. Specifically, the Time Capsule is a wireless router with a built in hard drive that can be used to store the backups of your Mac using the Time Machine application.

When the Time Capsule is connected to a modem with an Internet connection, it functions as a wired and wireless router in addition to acting as a storage device for backups. So, devices can connect to the Time Capsule using a wired Ethernet cable connection or a wireless connection.

If you want the Time Capsule to function as a router, connect an Ethernet cable from your modem to the WAN "O" port on the Time Capsule. Once that connection is established, then you can configure the Time Capsule as you wish.

The Time Capsule performs as a Level-2 switch between its LAN (opposing arrows) ports, but performs as a Level-3 router between any of these same LAN ports and the WAN (circle-of-dots) port when configured with Router Mode = DHCP and NAT.

In other words, it only performs as a router when network traffic comes in or goes out to the Internet from your local network, otherwise it performs as an Ethernet switch for all traffic between devices on that same local network.