There's that, but it also seems weird that Sauron would voluntarily become an addled hobo. There's nothing that ever suggested that in the lore, in fact quite the opposite - Sauron was obsessed with appearing confident and attractive, which is why it was such a blow to him no longer being able "to assume a fair form" after the fall of Númenor.
Whereas Gandalf INTENTIONALLY chose to appear as an old man, as a reminder of his own humility. It would also make sense that he'd be confused and disoriented, having come to Middle Earth for the very first time and indeed having assumed human form for the first time. Whereas Sauron has been in ME for literal ages.
I guess they could take into whatever direction they want, but I'd be VERY surprised if they changed things on that scale. Having Gandalf show up a little earlier isn't a big change - it's still the same reason, the same motivation, the same general everything except for the specific moment in time. If it was Sauron (or even Melkor) they'd have to change so many other things.