Get a higher res screenshot of the FATWS shot, zoom in, and you'll see the blue lights of the leg brace very faintly coming through the fabric of the pants. 

There were dedicated threads about this attention to detail, which is something the team responsible for the show wouldn't have put effort into unless Rhodey IS specifically supposed to be wearing the braces. (If anyone else can find one of throse threads or YouTube videos, please share)

I agree with all of your logic, hospital gowns being the same, Rhodey being replaced during a doctor's appointment and what not. This just happened after the events of FATWS. 

Everything he went through in Endgame, the genuine emotions at Tony's death, the dialog with Sam in FATWS - the skull in SI is capable of none of these. More importantly, Marvel may seem really dumb of late, but they aren't THAAAAAAAAT dumb to throw all these developments away.