This app was mentioned in 8 comments, with an average of 1.38 upvotes
I believe there is a newer, unofficial version of osudroid floating around somewhere that supports all the latest beatmaps. I don't want to link anything here in case it breaks a rule, so you'll have to look around. It's completely separate from the Google Play version. You'll know you have the right app if it has a similar interface to the PC version of osu but a bit more basic.
You can also try this game, it has its own osu mode and supports downloaded osu beatmaps.
I've always ran and enjoyed Beats - https://play.google.com/store/apps/details?id=com.beatsportable.beats&hl=en_US&gl=US which also uses simfiles (I use the same files as I use in Stepmania)
https://play.google.com/store/apps/details?id=com.beatsportable.beats Beats, Advanced Rhythm has been around since 2014 I think and works the same way (you DL songs/beatmaps from OSU then load it). It's pretty good
Any DanceDance Revolution player here? I started to play an Android app some months ago and I am starting to be able to survive songs with 13 levels of difficult. Since Youtube videos about this app are pretty noob I thought if it would be a nice idea to upload videos with my gameplays. For example today I did a Max Combo from a song with 9 levels of difficult. What do you guys think about it?
I love Beats. You can play osu! songs on it and it runs really well!
Have you tried Beats, Advanced Rhythm Game?
I don't see it on the Amazon app store so you'd have to sideload it, but it says "added basic Amazon Fire TV remote/controller support" in the description.
[EDIT]: Thought I'd link a couple of similar games that are on the amazon app store but don't have controller support (I think): Cytus and Music Hero
You actually had a very thorough list there, only ones I would add are: Beats, Advanced Rhythm Game (https://play.google.com/store/apps/details?id=com.beatsportable.beats) plays DDR, Osu, Stepmania maps
T-Aiko! (https://play.google.com/store/apps/details?id=guy.taiko)
Neon Rhythm (https://play.google.com/store/apps/details?id=com.bitflake.neonrhythm) use your own music, very minimal interface
Rhythm Samurai! (https://play.google.com/store/apps/details?id=jp.colopl.enzan)
Music Runner (https://play.google.com/store/apps/details?id=com.furthergames.soundescape)
Music Ride 2 (https://play.google.com/store/apps/details?id=com.zr.music.run) Audiosurf clone
and last but not least, SongArc (https://play.google.com/store/apps/details?id=com.songarc.songarc) play with your own songs, fun gameplay style that uses an arc you touch points across and device shakes for some
And of course Osu, also the PianoHolic games by Cherry, Magic Piano, and Walk Band/Perfect Piano by Revontulet Soft (although those are more tools, they do have a p2p play mode)
EDIT: Didn't think a DJMax game made it over to Android, thanks so much