Memes

I think you're imagining this as a more conscious process than it actually is. The reason you need to go out and research this is because these men aren't going around saying, "yeah, I was going to put my pepsi can in the metal recycling but I was worried my bro would call me gay so I just put it in the general waste." Instead, sometimes men put pepsi cans in general waste, and sometimes men do things due to social perceptions, and sometimes those social perceptions are that certain things are "unmanly" and working out which things are related to each other is quite hard.

So there's no paradox here. All people are subject to social pressures, and the vast majority of people make some effort to conform to those pressures in order to fit in and to receive approval from the people they value. Conforming to fit in isn't "weak" or "insecure", it's the nature of being a social animal, and is done instinctively - if you think it's done "obsessively" then you're imposing the analytical mindset of someone studying the evidence on the subjects of the research, which is a fundamental error. It'd be like saying someone who subconsciously mirrors the mannerisms of someone they respect is "obsessed" with getting their approval, when they likely don't realise they're doing it.

Also, I feel like they’ve never considered that gay doesn’t necessarily mean effeminate. Or even that effeminate doesn’t necessarily mean weak/meek.

They almost certainly haven't because, again, if you're "considering" it, it's not the right concept. The concept that people are trying to avoid is the one that's labeled "gay" by their peers, which is really more of a gender thing than a sexuality thing; "what are you, gay?" isn't a question about someone's sexuality, it's a suggestion that someone is not conforming to the gender role expected of them. You can't successfully challenge that by saying "ackshually gay people can be v strong and they forget to put the pepsi can in the correct bin far more often than you might imagine." They'll just reply with, "OK bro sounds pretty gay," because you didn't challenge them on what they meant, only on what you thought they meant.

The challenge has to be more along the lines of creating a better awareness of societal expectations, tolerance of people who don't conform to them, and building up positive associations between behaviours we want to promote and conforming things people already value, to help them see things in a new light.