Unfortunately, you are not going to like the answer........because Apple never designed the Time Capsule (TC)......using a WiFi connection...... to extend a WiFi network provided by a 3rd party router.
The TC will work as you want.......IF......the TC can remain connected to the Spectrum router using a permanent wired Ethernet cable connection.
If you cannot run the Ethernet cable from the TC location back to the Spectrum router, a pair of Ethernet over powerline adapters might be able to accomplish the same goal by using the existing AC wiring in your home.
I emphasized the word "might" to mean just that. Powerline works OK in some homes and not others due to the different ways in which homes are wired. Adapters from one manufacturer might work OK in one home, but not in another home, or not work at all. There is no one size fits all, or one model that works well in all applications.
An Ethernet cable is a much better solution than trying to use adapters, but sometimes you just can't run the cable.
Post back if you need more details. Hopefully, you will be able to run the Ethernet cable.
If you cannot run the cable, and need to locate the TC close to the Spectrum router, you can use the TC as a back up hard drive only. It won't provide WiFi service at all in this type of setup.