That is incredibly untrue. If anything it's the opposite. (obviously this is hyperbole but please hear out my amateur gay sociological research)
Whenever I hear a guy brag about his dick, it's always disappointing. At best it's average. Unless they're a porn star. I gather that with porn stars "a big dick" is more than what regular people consider big so if it's notable enough in that category it would be way too obvious to just ignore talking about.
But in real life, any guy with a lead pipe down there never seems to wanna brag about it. They will call it average or slightly bigger than average. I mean think of how many unassuming lanky skinny nerds or twinks you know that somehow have a tree trunk compared to the bodybuilder hunks. Part of it is just confidence that they don't have to care that much about how big they are. Their insecurities are in other areas of their life. They're not gonna be embarrassed if their sexual partner rips off their pants cause they already know it won't be disappointing. Unless I guess if their partner can't handle that much, people I call cowards.
Moreover bragging about a big dick always sounds incredibly douchey and makes you look like you're desperately compensating for something.