> "Working on math problems with students, giving study advice, and sometimes talking about my students' academic plans (e.g. recommending other classes for them to take, or telling them about resources they can access on campus)"

By your own words basically it's potentially kind of a social group where a subset of students get extra instruction, advice, and help, while maybe making a personal connection with their professor. I'm not saying that's evil. But it's definitely a resource that some are better able to leverage than others for numerous reasons. (Which also isn't inherently a bad thing, either.)

So while going to see your prof and asking for help is reasonable, it's not that simple, and viewing that as the extent of what office hours are is very reductive.

During my own time in college many profs emphasized that they absolutely don't give extensions, that you're all adults now, take responsibility, etc. I even had one math professor give the entire class a lecture (I quote) about "In computing we have RtFM, while here we have RtFT: Read the Fucking Textbook," because some the questions he was getting during class were a bit rough. And of course the textbook students were supposed to "RtFT" was also a bit rough and written by his own department so they could make money of forcing students to buy it. I think pretty much everyone in the class was terrified of going to his office hours after that lecture.

This isn't to say there aren't plenty of very decent professors. I once accidentally swapped two things on my calendar and missed a final. It was terrifying. I thought it was going to be a life altering disaster, but the prof let me take the final in their office. But up until the moment they did that and said it was okay, I pretty much thought the rest of my life was going to be severely complicated by a calendar screw-up.

But everything is really just a morass of YMMV, cross your fingers.

Maybe more to the point, I tend to remember an open door policy of some sort always being much more comfortable than office hours. It didn't feel like you'd have to reschedule work hours to make office hours, and it didn't feel like you were reaching by asking a prof to meet outside of office hours because you had another class during that time, etc.

And yes, many professors are happy to do things in a more ad-hoc way. It's just that a system that was supposed to be a simple way to make it easy to get help really isn't as straightforward or as helpful as it looks at first glance, and many students have a great deal of uncertainty about professor interactions since they can be extremely variable.

But no matter how things are done, there's a big ambiguity zone.

Is spending 40 minutes with your prof discussing a potential paper topic fine, or is it taking away from another student who is struggling with with something more foundational? What would happen if every student in the whole class tried to go to the same office hours for that? Are office hours only a thing that even "works" at all because only a select minority of students feel casual enough about them to take advantage? If a student thinks it's not right to get special help on something that's only ambiguously a problem are they at fault or not? If you want to learn more about something that was only mentioned in passing and not course related is going for that okay or not? If I'm at office hours trying to get help on a problem and the person in front of me is just arguing about why they should get 5 more points on the last exam is that alright?

I mean it's not like I hate office hours. I just think there one of many parts of college with good intentions that are in practice vastly more complicated, messy, and weird unless you come in very capable of being ...breezy.