Q: Bad ethernet port?

It's a long shot, but have you tried...

1. Resetting the SMC,

as per > Reset the System Management Controller (SMC) on your Mac - Apple Support

2. Exhausting everything Ethernet related,

at > Solutions for connecting to the Internet, setting up a small network, and troubleshooting - Apple Support

Clean Ethernet port with cotton swab could solve the problem.

If your ethernet port turns out to be dead you can get USB-ethernet and Thunderbolt-ethernet adaptors for a lot less than the cost of a repair.

C.

All the suggestions you have received are really good. I'd resort to a repair as a last resort because a repair would be extremely expensive. All the external ports on iMacs are part of the main logic board, a repair normally always means a replacement of that board.