I couldn't find the perfect app you're looking for - but here's as close as I could get and I couldn't write this in the comments because it's too long :D
I checked out the list in the comment and tried 3 apps -
Timbre: Easy to use and good for splitting videos and trimming, but I lacked the important feature (to me) to put music files on top of the video
FxGuru was definitely for effects mostly (I only watched the advertisement video), so not what I was looking for
FilmoraGo was what I personally was looking for: Join, trim videos and put music on top. You can edit pretty small steps (to decimals after a second), but not a single frame I think. Music fades out in the end automatically which was perfect in my case, but if you add transitions be careful cause it will shorten the overall video. Free, no advertising in the finished video or in the program (at least not while I used it). Too many permissions, but I can control individual permissions on my phone so I just blocked most of them and the app still worked :)
They had some (imo bad) themes but fortunately you don't have to use them and some of the transition effects I didn't find too bad. I don't think on a small 5inch android phone screen like mine one can expect something even better unless they're willing to sacrifice user experience and have much more time to kill. But of course I'd be glad if someone can prove me wrong :)
EDIT: I use YouCut nowadays, I like it even more than FilmoraGo
It's quite minimalistic but enough for me (cut/trim/join/crop/music) with a clean user experience.