You think Mathematicians in general have people skills? Most Mathematicians i met were good people but they definitely kinda lived in their own world, they were super helpful, but only if you explicitly asked for their help.
I studied at the VU, where Andy Tanenbaum, writer of a ton of famous CS books teaches. Of course I followed his classes. His books are excellent, but his classes are basically him reciting his books from memory. The jokes are literally identical.
At some point, more and more of his classes would be taught by Maarten van Steen instead; he's an excellent teacher who really knows how to engage the students and make them think about problems, instead of merely telling them the answer.
There was another professor who, as far as I know, didn't do any research at all; he just taught a ton of classes.
A professor who is great at one thing is not necessarily good at something else.
I don't think that that is an introductory course.
As an example: you'd reasonably expect a student with mathematical maturity to know what a set, the Cartesian product of two sets, and a function are. That's high school level math.
From there you can define a ring and a module over a ring, in like 5 mins.
So at that point the Prof, strictly speaking, told you everything you need to know to start studying modules.
Of course, many students will be like whaaaaaa?! When a ring is defined. But here's the point: Mathematical maturity does not mean that you already know what a ring is, but when the Prof defines a ring, you are following along.
Anyway, my point is that there is nothing specific that you need to know.
Any good definition of mathematical maturity would likely be so technical that you need mathematical maturity to understand it.
Some people are better with math than others. Undergrad math majors thought the major was pretty easy. Many of us in engineering majors--who weren't that bad at math (it was a good school)--could no more have graduated with a degree in math (or physics) than have flapped our arms and flown.
Most colleges have classes designed to foster mathatical maturity, but that is done by how the class is taught, not what is taught in it.
It didn't help that I went to uni at 24 IIRC while I was working my ass off, so I wasn't that happy to put up with the BS.
Calling it gatekeeping seems like a fish complaining that the birds are gatekeeping the sky. The sky is open for anyone, but only if you have wings to fly.
It also usually helps if it's an interesting problem...