Hi. Welcome to the forum.
It sounds like the TiVo THINKS it's changing the channels on the box, but it isn't and the box is staying on channel 3, feeding that and only that to the TiVo.
I'd make a suggestion on how to wire things up, but you haven't told us if you're getting any analog channels. (Can the TiVo or even an old analog TV tune ANYTHING without the cable box, cable feeding directly from the wall into the TV or TiVo?)
We need to know if you still have any analog channels on your cable system.
Let us know, and that'll be a good starting point so we can try to help you check or redo your wiring.
No sidewalks with beef, please.