Thanks for all the replies guys! After reading the links Tony provided, I went with the "Listen" app. This is already an improvement over the standard music app, but it still uses the same decoder, so the files still have the same errors. Though I can imagine that's because Apple doesn't allow for other decoders to be used, same as they require every competing browser app to use the Safari engine.
That said, this is already a big improvement for in-car use, since it uses gestures exclusively and not button presses. It doesn't have any of the smart Empeg features though.

I very much hope for somebody to write an Empeg like app. I would pay money for that. It's a shame that Raspberry Pi Empeg project never was released.