Now, I only watched through the end of S10, but on the show I watched, Sam had a very good idea of just what Dean went through, and what Dean had done for him. True - he didn't see it at the beginning. In S1, Sam grumbled, "Dad never treated you like that. You were perfect. He was all over my case."
But ten episodes later, Sam and Dean fought a shtriga and Sam came face-to-face with the reality of his brother's upbringing. From then on, Sam was repeatedly confronted with the difficulties Dean experienced growing up. Something Wicked, Dark Side of the Moon, and Bad Boys deal most explicitly with these issues. And Sam learns from these experiences: "You know, I've really given you a lot of crap, for always following Dad's orders," he says in Something Wicked. "But I know why you do it." In DSotM, he tells his brother, “I just never realized how long you’ve been cleaning up Dad’s messes."
Sam certainly doesn't downplay what Dean has done for him. "You sacrifice everything for me," he told Dean at the end of S2. Near the end of S10, he told Castiel that he owed Dean everything.
If one of John Winchester's sons has an idealized vision of what his brother's childhood was like, it's Dean. In Dean's dream in DaLDoM, Dean's dream-self claims that John doted on Sam, and that he loved Sam (implying he didn't love Dean). This of course is immediately contradicted a few minutes later in the dream by Dean insisting that John was never there for Sam, but there is no hint that Dean notices the contradiction. As far as I can tell, Dean in Brother's Keeper still believes his father doted upon his brother.
Even when Dean acknowledges that John wasn't there for Sam, he makes another inaccurate statement: Claiming he, Dean, was always there for Sam. In fact, flashback episodes and discussions of their childhood give us plenty of examples of Dean not being there for Sam.
The problem here stems from Dean's warped evaluation of his own ability to parent Sam. John's expectation that Dean take on some of the parenting of Sam's was unrealistic to the point of being abusive - towards both of his son. A traumatized child cannot parent another child only four years younger.
But Dean usually seems to look back on his caretaker role as a success. When he does acknowledge mistakes (Something Wicked, PPMM), he takes responsibility for those mistakes as if he had made them as an adult. Dean's inability to see his attempt to parent Sam as a child's endeavor is harmful both to himself (since he places unrealistic weight and expectations on himself, just as his father placed on him), and to Sam (because as long as the idea that Dean was capable of being a good parent is a reasonable one, there's no way that Dean can recognize how difficult it was for Sam to grow up in his care).
On a Doylist level, it's also true that the viewers are forced to acknowledge that Sam's initial view of Dean's role in the family was wrong, while Dean's accurate view of Sam is never explicitly challenged. Something Wicked and Bad Boys explicitly repudiate the idea that John saw Dean as "perfect". Whereas the idea of Sam as doted upon, though inconsistent with what we know, is never explicitly addressed after DaLDoM. And I think this is why so many fans continued to hold the belief that it was true.