With Internet radio, (streaming) do AM\FM tuners serve any purpose in HEA?

If you have radio stations that are worth listening to then sure.    

It depends. Classic great tuners can be found for <$200 now. Some broadcasters down rez so much that their over the air broadcasts might be better. Some major cities have frequently schedule (almost weekly) live (!) opera. For fans of this a good tuner , some decent interconnects and a well placed (!) antenna can be a very, very worthwhile investment.

Internet radio has not rendered good tuners obsolete, imo. A first-class FM signal will walk all over the typically digitally compressed streamed radio station, and this can be especially noticeable on live broadcasts. Of course, most FM stations are about as poorly compressed as much of today's pop music. Or worse.