The TV can receive encrypted video. A lot of capture cards don't, on purpose, because people will use them to steal stuff. Generally, if an output can't be encrypted (SDI digital, or Composite analog, or whatever), then a device is supposed to disable that output when it receives an encrypted input.
If your cable box is encrypted (as I would expect, given that it pretty much only plays commercially copyrighted content), then that's why the TV works and the capture doesn't. Sounds like both your capture card and your converter are doing exactly as they're supposed to.
Again, OBS has nothing to do with it, and we're not going to help you circumvent the copy protection that's intentionally keeping it from working.