I don't think he does. Without being stricken with polio, Roosevelt probably doesn't take a break from politics, and thus, probably runs for the Senate or Governorship much earlier that OTL. If he goes for the Senate, that gives him a voting record to run against, and, that, combined with the fact that Senators very seldom get elected President, plays against him, in my mind.
If he does get elected Governor earlier, I think that makes him 'old news' politically a lot faster, and he probably gets the Democratic presidential nomination earlier than OTL as well. IIRC, his speech at the 1924 Democratic National Convention (in which he nominated Smith) was so good that the delegates, had he not been so afflicted with polio and still in the early stages of recovery, considered nominating Roosevelt right then and there instead.
If Roosevelt does manage to become President, he's not going to be the same Roosevelt that we know from OTL, either. Roosevelt's affliction with polio changed the way he viewed the world and took him from being a Wilsonian liberal to the social democrat that he governed as, galvanizing Roosevelt toward the concerns of the disadvantaged and the out of luck.