I grew up reading 'chick fantasy' that my mom had lying around (Andre Norton, Anne McCaffrey, Patricia McKillip, Linda Bushyager) as well as some various sci-fi/fantasy hybrid stuff like Fred Saberhagen and Roger Zelazny, and reading comic books, where mind-reading, hypnotizing people, throwing stuff around 'telekinetically,' astral projecting, etc. were the very definitions of 'magic.'
Some (Andre Norton, in particular) mixed and matched, others didn't even bother, having basic psi stuff being just generic magic, as it was for the couple of thousand years it was before the word 'psionic' was coined.
Call ESP 'the sight' or 'the third eye' and psychic domination 'glamour' or 'enchantment' and TK 'poltergeist' or 'unseen hands' or whatever and it's all magic by another name. Half of TV 'magic' is telekinetic stuff, it seems, from Willow Rosenberg pointing and stuff flying around to BBC Merlin pointing and stuff flying around to Zeddicus Z'ul Zorander pointing and stuff either flying around or bursting into flames...
The only quibble I have with psionics in D&D is that about 85% of the stuff one would do with psionics has already been shamelessly poached by the spellcasting system anyway.
When anyone does use psionics in D&D, we use psionics / magic transparency, and just flavor it as a different way to manipulate supernatural forces, rather than deal with quasi-compatibility issues, so Jean Grey becomes Dark Willow anyway.