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

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.

I listen to FM stereo at least 4 hours a day on my office system.  Thankfully, there are still 3 excellent Rock stations in my area.  

I have several nice tuners in my collection - including a Scott tube tuner and a McIntosh MR77. Presently listening to NPR "All Things Considered" on 89.7 FM (WUSF Tampa) - using the Diatone (Mitsubishi) DA-F10 (1978). Excellent SQ! This is the only station I listen to because all the commercial stations in my area sound like dreck - musically and sonically. WUSF is a college station. Every night they have an excellent Jazz program from 9 PM to 5 AM! I am blessed to be able to listen to great Jazz programming every night in excellent sound!

So if you are within range of a college station you are indeed fortunate!