At schools my DC have attended, dads have turned up to assemblies, parties, sports days and whatnot. Usually there are more mothers than fathers, but definitely men do "show up".
But what they DON'T do, in my experience, is be the default parent on the FB group, or the email list, for the less critical but still important announcements or requests about cake sales, yoghurt pots for cress heads, lost property "has anyone seen Jimmy's hat" mails, or "what time does the coach leave for the trip" sort of admin.
They'll be there when they see a visible benefit and will usually - for sports days, soft play parties etc - have got the relevant info from their other half. They probably didn't source the yellow T shirt for sports day, answer the email about contributing to the class present for the party (or RSVP to the invite in the first place - & it's always mothers mobiles to RSVP to, regardless of who works full time as far as I can see), or make sure the kid had the costume in for the concert.
So it's not just "who signs up". Who signs up takes responsibility for being the person to manage the things that crop up from that email list/FB group.
The mother can forward emails to their OH and tell him about yoghurt pots or yellow t shirts if they want to delegate, of course. But if the hypothetical couple both work full time, why is the woman managing and delegating and bearing the responsibility? Why didn't the OH sign up to delegate instead? Of those 30 kids on that optional list, why is the mother the one who has put their hand up to that responsibility to know what's going on in the DC life? Why can't BOTH sign up-voluntarily- to get the emails, and then the couple had a hypothetical discussion "did you see about the yellow t shirt, DW, I'll pick one up from Tesco on the way home if we don't have one for DS?" Instead of "DH can you pick up..." where it's just glorified mental management.
I honestly honestly can't see why this is controversial to anyone - that both parents should be equally interested in the minutiae of their DCs lives. It's hardly duplicating effort is it, if everyone's communication is so fucking brilliant anyway.