I don't think the stigmatisation of men showing emotions is exclusively Anglophone culture. I live in Ireland and there isn't really a stigma of men showing emotions because of public awareness campaign about mental health for both men and women. But like you said, I've met couple of overly manly men jerks and uncaring women, but they're the ones not worth your time and in tiny minority.

In any case, some cultures have antiquated machismo mindset which is sporadic across the world.

I am from Spain. When I open to my ex when we were in the process of end our relationship she told me to stop to victimize myself. I think that the relationship started to fell down when I started to be myself in front of her (expressed doubts, weakness, expressing enjoy for things...).

As a Portuguese (that has also lived in a few other countries in Europe) I would say that it's more that there is a range of emotions that men can express without that being frowned upon were certain things are still frowned upon if you show them openly (mainly around sadness) though for example openly showing tenderness for your partner or children is expected and even approved (unlike certain other cultures were men are expect to be distant).

Mind you, in some cultures the limits on expression of emotions or selectivity about which emotions you are expected to express is pretty high for both men and women (for example, the Dutch in general tend to refrain from expressing much emotion to strangers) and in some cases there is even such a strong expectation that you react in certain ways that it leads to people in general faking expressions of emotion (the English upper and upper middle classes are pretty big on showing the "appropriate" reaction independently of feeling it).

I would say (from contact with Americans and consuming some American media as well as having lived in England) that the expectations on what emotions people should be expressing are quite different and in England they're even very much defined by people's social class (for example, the "English Gentleman" is entirely a façade - all about what you show, not at all about what you think - and occupies the same place in terms of male behaviour expectations for traditional old-money upper class English men as the bossy slightly-angry assertive go-gotter seems to occupy in the US).

So far I generally have seen a tendency for frowning upon grown up men expressing sadness for themselves (though in some countries, not for expressing sadness in empathy with others and their pain, especially if they're close family) and have also noticed equivalent expectations on the expression of emotion by women (for example, it seems to me that middle and upper class English women have a massive weight of social expectations on them in terms of what they're expect to show to others - including the emotions they express - in lots of situations, and a lot of it is about reacting with the "appropriate" emotion in some situations even if they don't feel it)

I have noticed a pattern in myself, which I suspect could be true for others too. Namely, that I am much more likely to care about someone if the feeling is mutual. On a rational level, I can recognize that this or that person has had a bad time and deserve better, and I want to help them, but if they are hostile or indifferent to me, I kind of stop caring. And vice versa, if I feel that someone cares about me, I will care a lot about them. If this is the case for two people, it can quickly spiral either towards more mutual caring, or more mutual indifference.
Now here's the tricky part, how do we influence this trajectory? The only way I can think of is to care even if the other person isn't caring back. Polarizing language can feel good, to assert yourself when you feel hurt, but... is it helpful or detrimental for the bigger picture? It's so circular and self-reinforcing. So hard to escape.
I get the feeling that many people argue that "well, when they start treating me right, then I will start treating them right, but until then I don't care", and sure, I understand that feeling. But the feeling is probably mutual.