Controlling and being in charge of the texture is one of the things that makes piano playing so satisfying. We are one-person orchestras (if I may exaggerate a bit)
In my teaching, balance between the hands gets addressed when we have a piece with a tune in the Left Hand. Trinity Grade 2 had a fabulous exercise a few years ago called Manatee Parade.
At a higher level, we talk about it for the Satie Gymnopedie No 1, where we want a rich bass, quieter middle chords and the right level for the melody above.
Here’s a harder example from Prokofiev’s Romeo and Juliet.
And finally an example when it’s all within the one hand, Scriabin Nocturne for the left hand- the sighing melody, the harmony underneath and the chromatic ascent above which I wanted to call a descant – but that might conjure up sounds of a hymn where the descant may be the loudest part which is not what we want here.
I find balancing such textures and paying attention to voice leading the hardest thing in piano playing (I’m excluding virtuosic semiquavers and large leaps as I don’t play at that level). Yet it is immensely satisfying.