I played it on PC for the last year or two using the XB1 Controller with no problems thanks to the ability to map combinations of buttons.

Gamestop had PS4 version of Elite on sale for $20 so I picked it up there a week back and it plays great on PS4 as well (looks like it's still on sale for $20 now as I type this):
https://www.gamestop.com/ps4/games/elite-dangerous-legendary-edition/148662

I'm lucky enough to own a PS4 and a PS4Pro, and on PS4Pro the framerate and visuals look like they do on the PC, but on the original PS4 there are some slowdowns compared to PC.

Either way it's really playable with the controller, I own multiple HOTAS (X52, T16000.M) but the XB1 controller and now DS4 controller are my favorite way to play and I can dogfight with the best using controller.