IMO knowing about sharps and flats is an impediment to transposition. I play alto recorder, and I was very careful to avoid learning any sheet music. Instead I wrote a simple Python script to convert MIDI files to numbers, with 0 representing the lowest note, 1 representing a semitone above it, and so on. I then memorized the fingering for each number. I can transpose in my head just by adding or subtracting. And I practice scales based on the semitone intervals between the degrees of the scale, so I only need to know major/minor/wholetone/pentatonic etc. and just change the starting note.

This is more difficult on the standard piano because of the badly designed keyboard, but you can get isomorphic keyboards that let you play all the keys without learning all the sharps and flats.