You could, but would you - or most average FM listeners - bother?
Does HD-FM really sound that much better than analog? I just have never heard any "typical listener" complain that they would listen to terrestrial radio more "if broadcasters could just get the sound quality better." The complaints I hear is that the programming is uninteresting.
Analog FM provides good dynamic range and a reasonable degree of noise immunity, certainly comparable with what you can get from mp3 files. If the FM signal isn't overprocessed (no guarantee THAT won't be a continued problem with HD) it sounds fine for any reasonable listening application. So the poor building penetration, insufficient coverage and mode-hopping that HD-capable receivers produce are a poor tradeoff for the dubious benefit of listening "in digital" as opposed to analog.
As far as HD-AM is concerned: yecchh with that codec. Even assuming you can get a reasonably reliable digital lock. I'd rather deal with analog noise on AM, thanks.
HD: a solution in search of a problem, that brings along a whole wheelbarrow of new, highly objectionable problems not present with analog broadcasting.