I have at least tree stars on all All Stars songs, but I didn't get the promised trophy ("All Stars Fan").
I did something like this (not 100% sure):
1. Played a few AS songs, minimum three stars on all.
2. Played all AS songs a couple of weeks later, minimum three stars on all except one, where I got two stars.
3. Played the two star song again and got more than three stars.
But no trophy.
There is a similar issue with JD 2019, where I got at least three stars on all regular songs, but not the promised trophy ("Out of This World!"). I managed to get the same trophy i 2020, though.
Another minor issue is that the trophy list isn't updated when I get a trophy and switch to the PS4 main page with the Playstation button to browse through trophies. It's updated later, at least after a reboot. I can live with that.
Well, now I got the trophy. This is what happened:
All stars mode showed at least three stars on all songs, but I didn't get a trophy.
I had superstar on High Hopes.
I played High Hopes again in All stars mode, got superstar and then suddenly the trophy.
My guess is that you have to get the stars when in All stars mode, and that the stars you see there may not count if you didn't get the stars in All stars mode. It could be like this for High Hopes, which is special because it's part of 2020, or it could be like this for all songs (I have JD Unlimited).
This must be a bug, and you should hopefully have enough info now to fix it.