I mean, I can understand him not wanting to be called Stretch Guy, but he seemed to quickly grant himself a grand name which implied leadership of the FF!
Someone should have called foul and retorted that it should be Mr. Elastic.
I think in Stan Lee's mind, Reed was the father-figure, and therefore the assumed leader. I think the name is an extension of that "well, duh, of course the oldest male is the leader" presumption on Stan Lee's part. It fairly reeks of a whole lot of problematic assumptions on Stan Lee's part, a whole host of problematic thoughts that were so "well, duh, of course" in his mind that it didn't even occur to him to spell it out in the story.