Setting aside legal/licensing issues, retrieving an MP2/MP4 (depending where in the world you are) stream via SDR would require massively more power both in terms of transferring the raw data to the CPU and processing to extract the signal from it. It also requires better hardware (IIRC ATSC, for example, requires >10-bit ADC while FM radio is fine with an 8-bit ADC.) So a SDR-based TV tuner would drain your phone battery quite quickly and still not work as well as a dedicated TV tuner streamed to your phone.

If that weren't enough, there's also the fact that the digital TV signals don't work well with moving antennas on the receiving end...

Maybe actually. In some jurisdictions this would require a license from the government to do the reception. The UK is like this with their tv licenses for the BBC and similar. I can't imagine theyre the only ones like this. And then there's possibly compression technologies that might need to be licensed from the mpeg group, etc.

Some phones in Japan did (and maybe still do) have TV tuners even in the pre-smartphone days.