There's an English jingle that I was taught (1970s), and my child also:
Thirty days hath September, April, June and November.
All the rest have thirty one, save February alone
Which has twenty eight days clear, and twenty nine each leap year.
(edit: in Australia, for reference)
I think this might be a cultural thing. A friend of mine from Taiwan was perplexed I didn't have this memorized, my reaction was the same as yours.
Why is this constraint necessary? If you can even so much as think about it or count knuckles (let alone that you happen to have memorized the number of days per month), then this calendar is useful.
Doing the knuckle dance every time you glance at it, to know whether to change the month already, seems annoying. (Yeah you'd memorize it for that month after a bit, but still you have to pay attention.)
What's wrong with counting on knuckles? You adjust your calendar with this contraption once per month.
And yet it's true. It would take me a lot of thinking to answer how many days are in an arbitrary month. And I'd probably only be 80% confident. I truly don't understand how I don't know this either. But I've never made an effort to learn it, nor has it been a problem.
Doing the knuckle dance every time you glance at it, to know whether to change the month already, seems annoying. (Yeah you'd memorize it for that month after a bit, but still you have to pay attention.)
Please straight up ask ten people, even if your circles are on average highly educated, and see how many know an arbitrary month (not significant ones like February, or December where the 31st is new year's) without having to think about it or count knuckles. Asking about the current or an adjacent month might also be something they remember from recently seeing calendars that tell you this information.
I think it's very uncommon to have this memorized to the point where using a calendar that doesn't tell you is as easy as using one that does.