Action anime tend to have a lot of action, but anime is a medium and not a genre, so there is a…

Action anime tend to have a lot of action, but anime is a medium and not a genre, so there is a huge amount of diversity in content: popular genres right now are high fantasy parody, romance-comedy, and supernatural police procedurals. One thing anime *does* have in terms of content across genres is an emphasis on emotion, in part because one of the handful of things animation (and particularly anime) is really good at is showing the way people feel.

Cowboy Bebop is a fantastic show, and very interesting. You should watch it, in part because it specifically grapples with the topics you cover on Humungus: how internalized ideas about masculinity can be limiting and affect the whole of society. It's also a great found-family story, and has rip-roaring action and sci-fi and mystery in every episode. But, you won't find a lot of open displays of emotion in it, in part because it's specifically about people who feel bound by social pressure and acculturation to mute their emotional displays or redirect them into violence.

If you're specifically looking for loveable emotional characters & good action, try My Hero Academia, which is about a world that (in response to widespread mutations) has developed government institutions and training programs for superheros along with media apparatuses. Basically, imagine if The Boys was less cynical.