I would have thought the opposite: that what distinguishes bromances from buddy pictures (which have been around for a long time in American movies) is precisely the degree of hysteria they exhibit concerning homosexuality.
There are probably exceptions, cases where the hysteria is mixed with more nuanced moments; and I also admit that there have recently been a lot of bromances that I haven't seen. But I use the term 'hysteria' advisedly. Isn't the core of the bromance, insofar as it's a bromance and not just a buddy picture, the awkward moment when the friends have to negotiate around the possibility of love? When the relation between male bodies onscreen gets invested with the same kind of narrative affect that, in a romantic comedy, would be produced by dramatic irony? Isn't the outcome always a feeling of relief that this has been negotiated in a nonsexual way?
In a buddy picture, we might say that the fundamental relation is "homosocial": when women do show up, the men's relations to them actually develop and cement the relationship between the men. This structure resembles that of the men's clubs that you've mentioned. Doesn't the bromance turn this structure into a nervous joke? Even in movies like The Wedding Crashers that avoid showing the kind of dangerous physical intimacy I described a moment ago, the sexual relations to women have to be heightened beyond anything in a buddy picture: you need to actually show the sex scenes, even though the movie isn't really interested in the women. Where the classic homosocial relationship can't be absolutely disentangled from homosexuality, I suggest that the by speaking explicitly about homosexuality, the bromance develops a narrative necessity to make the separation final.
It seems true that bromances rely on the possibility of talking openly about homosexuality, or what you've called a greater level of acceptance in popular culture. But I think you've made two further assumptions that might be false: first, that this is an acceptance of the gay community, and second, that the movies not only rely on the possibility of explicitly speaking about homosexuality, but are themselves in some sense "open" to it. The first point is tangential, but I would only suggest that if a community is represented at all in these movies (usually in comic peripheral characters), this could be a way of taming homosexuality. The second point is more central, and I've already suggested that there is something akin to the idea of "gay panic" in these movies. Because this notion itself can now be spoken, we sometimes get a few levels of pseudo-reflection: the characters themselves are aware of the possibility that they are panicking. In the movies that I've seen, this has not led to a better understanding of men's relations to each other, but to a higher order of panic -- again, the anxious joke at the core of the bromance.
There was recently a movie called Humpday that tried to use the bromance formula to treat these questions more seriously. I'm not sure it was very successful; but if you haven't seen it, you might still find it a worthwhile counterpart to the less conscious efforts of most bromance films.