My time capsule will not connect to my wifi network

I am trying to connect my 3TB time capsule to my wifi home network. I want to use the TC wirelessly without ethernet connection.

You can do this.....IF.....your main router is also another Apple AirPort router. The Time Capsule will use a a setting of "Extend a wireless network" when it is set up this way. But, this setting will not work with a 3rd party WiFi network.

I am using a Linksys Velop system.

Unfortunately, you will need to change your plans to connect the Time Capsule to your network. The Time Capsule cannot "extend" the Linksys network, and there is no "Join a wireless network" setting for the Time Capsule.

When I ask the Time Capsule to add to an existing network, it does not apparently see my network and gives me the option of one network which happens to be in a neighbor's home.

This tells you that your neighbor has an Apple router. The Time Capsule could connect to this network if you know the password for your neighbor's WiFi network.

All that you need to do is connect the Time Capsule to one of the Linksys mesh units using a permanent wired Ethernet cable, set up the Time Capsule to provide a WiFi network and then turn off the WiFi on the Time Capsule.

If you do this.......then.....you will be able to back up to the Time Capsule over the Linksys mesh network. If you don't want to see the Time Capsule, hide it somewhere near one of the mesh units. just remember to give it good ventilation.