Not just no, hell no. It's not obsolete.
It doesn't have the full-time service reach or sound quality of FM or streaming, but AM has one major benefit that FM doesn't: range.
50,000 watts at night on an AM station in the standard band will get you covering almost half of the country. A little higher frequencies than that, you can ride the skywave just about around the world. 50,000 on FM? Might get you the most important suburbs in your metro area.
And in the daytime, you use those frequencies for local service.
The stations I listen to on a regular basis are, for the most part, AM and shortwave. AM 650 WSM is my nighttime country station, when I want to hear the Opry; during the day, that falls on a little family-run AM station, AM 1300 WXRL. For the oldies, my go-to is shortwave station
5050 WRMI; if I don't have my shortwave, I can go to AM 740 CFZM out of Toronto, which is in range day or night. For standards, it's another local station, AM 1340 WKSN. That's not to mention scanning the band looking for live sports.
Is it all wine and roses? No. It's been a few years since my old go-to, AM 1590 WGGO, has played music. Now it's in the hands of a Catholic broadcaster. Another local station, AM 1330 WSPQ, went out of business a few years ago.
Someday I hope to own an AM station. With streaming as a supplement, I truly think it's still viable, and with AM licenses coming down in value to the point where it could very well be within reach soon enough... a guy can dream.