Setting up AirPort Time Capsule as a backup disk

When you see the option "Other AirPort Time Capsule" with the "Set Up" button but nothing happens when you click it, this usually means your Mac is not properly communicating with the Time Capsule (TC) through the AirPort Utility. First, open AirPort Utility manually and see if your TC shows up there. If it does, click on it and configure it directly—make sure it's fully integrated into your network (joined or extending the existing Wi-Fi, or bridged if using Ethernet). Assign it a static IP or reserve one via DHCP if you’re using manual IPs elsewhere.

Once it’s properly configured and online in AirPort Utility, open System Preferences > Time Machine, click “Select Backup Disk,” and it should now show your Time Capsule's disk by name—not just as “Other.” Select it and enter the password if prompted.

If it still doesn’t appear, try the following quick checklist:

• Make sure your iMac and TC are on the same subnet.
• Ensure that file sharing is enabled in the TC's settings.
• Reboot both the iMac and the TC.
• If you're connecting over Wi-Fi, try an Ethernet connection just for initial setup to rule out multicast discovery issues.

I linked up my AirPort Time Capsule to my network

If you did this, then the name of the Time Capsule will appear on the left panel of any Finder window under the "Locations" heading.

Do you see your Time Capsule's name here? If not, then the Time Capsule is not linked to your network and we'll need to take more steps to accomplish that goal.

Here I am assuming that your Time Capsule is connected to the modem/router using a permanent wired Ethernet cable connection......correct?