Nope. No playlists button at the bottom of my app. I tapped the music app, then at the bottom I see 'For you', 'New', 'Radio', 'Connect' and 'My Music'.
I think I'm supposed to tap 'My Music'. So I go ahead, and do that. Nope, still no playlists button... (Scratch my head, trying to remember where it is...). I'm currently looking at the last track I played, but I still can't remember where that playlists button is. Ok, let's hit the 'Back' button and see if I can find it.... Nope, now I have 'All Apple Music' or 'My Music'. Ah, just realised it remembered what I was searching for last time (because I couldn't find it in the first place in the messed up interface). Ok, clear the search... Aha, finally found the list of my playlists...
I'll forget this totally unintuitive, messed-up sequence that I seem to have to follow every time. So ok, maybe it doesn't take me ages, but I shouldn't have to go hunting for it, every time I attempt to use it.
I found out how to replace the CONNECT button with PLAYLISTS:
https://www.macrumors.com/2015/07/01/replace-connect-with-playlists-apple-music/
I must have done that before. Try it and see if it works. I hope this helps make it a bit easier