Almost all of us start from some place of privilege in some way – Gendry may be lower class than Tyrion, but Tyrion has a disability and Gendry doesn’t – so Jon Snow certainly has class advantages that many others in the Night’s Watch don’t, but he’s definitely below LC Jeor Mormont or Ser Alliser Thorne or Ser Denys Mallister
Moreover, social mobility doesn’t just apply to the very bottom and the very top; it’s a spectrum that goes all the way up (and all the way down, when we’re talking about downward mobility). So Jon Snow starts out as a bastard who will never inherit land nor title and becomes Lord Commander of the Night’s Watch and will probably end up as (one of) the savior(s) of humanity. Hard to top that.
As for Gendry, he starts as a bastard orphan, becomes an apprentice blacksmith, and is now a knight and has his own shop. That’s astonishing levels of social mobility for his society.