I cannot give you a concrete answer how to solve your problem. All I can I give you is my experience.
I had the same problem when I first got my HomePod, I didn't describe to Apple Music.
After about a week, I decided to start my Apple Music subscription.
Didn't have any problem after that.
I don't think you have to subscribe to Apple Music to physically play your playlists.
However, I believe, to take advantage of voice (Siri) commands you must subscribe.
Otherwise, I think you will have to physically play your playlists from your Mac, iPhone or iPad and then select HomePod as your output device.
HomePod Siri will play specific songs, but to recognize playlists, I believe you will have to subscribe to Apple Music.
That was my experience.
Now before you blame Apple, this does make perfect sense, if Apple is strategically using HomePod to enhance it's music service. I believe Apple is trying to give benefits to Apple Music subscribers as opposed to giving HomePod Siri benefits to everyone or competitors, for free. Which makes sense, if they are trying to grow the revenue for their 'services.'
Hope that helps.
-geekmee