Can I leave TV provider blank on my iPhone without affecting streaming apps through Apple TV?

What. happened when you tried?

I don't have one selected (because I don't have one) and I've notice no issues. If you have apps that require a cable subscription to work (they are, fortunately, getting fewer and fewer), I believe you'd have to select it for every app. Because I don't have cable, I don't have any of those apps so I can't test them.

sure, I never added a tv provider because I never had a tv provider