So I'll prefix with a certain level of TLDR since that's a big wall of text that online forms like this aren't particularly conducive to.
As a broad take though, I see two frameworks for men's 'rights' though.
There are the sorts clamoring for a reversion to old ways of patriarchal orders, who look to see things in this working man supporting a household and holding a veto-wielding authority over social and domestic order as a golden era perfection. This form is dead, regressive, and a vestige of an imagined past that will not come back.
There's another side though that fights against the still common perceptions that men must adhere to certain standards of behavior. The self-reliant, stoic, resilient sort that if they ask for help are considered failed in some fundamental way. This is speaking from an American standpoint since I have little context for other places cultural norms, but despite being publicly said (and in many places honestly held) that these norms are not the case there's still an underlying current that reinforces these standards. These challenges are quite relevant and alive today.
It's something of a multi-generational battle to alter what's expected of men in the modern world. That's the kind of men's rights that we shouldn't be so quick to dismiss as an irrelevant.