Regardless of the truism being true or not (I don't think it's true, but YMMV), I don't think it applies here because OP is interested in a specific person, not just seeking companionship in general. If they were ever going to get together, at least one of them would have to ask the other out.
I guess the 'problem' is that I don't expect anything to happen and therefore the whole process feels like a distraction, which is kind of the opposite of forfeiting all control and reason to it.
This seems like a contradiction to me! From the introductory post you sound pretty head-over-heels infatuated. Maybe you don't expect a relationship to arise out of it, but you're definitely heavily invested in it. It doesn't sound to me like you want it to go away, either? I apologise if I'm misreading your posts, but it seems you're interested in how to pursue this, or at least things to consider when considering whether to pursue or drop it.
The information you've given is pretty generic—the fact you're in constant proximity means regular opportunities to interact and bond, and develop that into a more extracurricular friendship and/or ask her out. But it's impossible to really read what's in her head or even in yours from what you've written, I guess? Not a qualified psychic. So, from what you have written:
I'm just sort of attempting to play it cool rn as I get to know her better
This is a decent general approach, depending on what playing it cool means. You want to get more comfortable with each other naturally, not distance yourself or play hard to get, not go overboard. Showing sincere interest in her as a person isn't a bad thing here, it's just about keeping things proportionate to the time you've known each other and the situations in which you're interacting, and being mutually comfortable. I'm not telling you to go up and compliment her appearance every day, nor stalk her to her dorms, but it's perfectly okay to tell someone when the opportunity arises they're cool or you want to get to know her better, or thanks for hanging out because it was fun and you enjoy her company (examples of comments that might be appropriate depending on casualness and amount of people in situation), because you do. The kind of grounds you're on dictates the boundaries, but in general the only way to get closer to someone is to try and see if they're interested too, and that applies as much for developing (emphasis: developing) an acquaintance into a friendship as it does into a relationship. It's hard to be affectionate and know how much affection is okay, and again very much depends on the person, but mutual affection (and this is anything from laughing and smiling to compliments/gestures of caring as well as physical affection: personal comfort on both your parts) promotes closeness.
And the thing is, you can put in as much effort as you like (and you should put in effort!), but nothing will happen unless she puts in effort too. But the more you get to know each other and the more it's evident she's comfortable around you and the more (hopefully) you hang out outside of class, etc., the more evident it will be whether she likes spending time with you or not.
As a person with severe social anxiety who left HS because of harassment—picking up on those types of social cues implicitly is terrifying, especially when you're primarily initiating. It opens up potential for insecurity and overanalysis: does she really even want to spend a second of the day with me, etc. Honestly, you can pretty much take it at face value, you learn to trust again. If someone wants to see you, they'll express it in the way natural to them. But the only way you can learn to do it is by experiencing those situations, interacting with more people, getting to know people as very variable so that you will learn to implicitly set a balance between adapting to get along with them and holding your ground as a person. Learned behaviour is based off learned experiences. Similarly, learned experiences can be subverted (I'm in therapy for this, lol). I understand from what you've posted you're growing into your own a little at your college as you test the waters, and that's good.
Some things are gonna suck, probably, but try to consciously internalise that college isn't inherently the same as high school, the people at your college aren't necessarily the same as at high school, you aren't the same person as you were then even maybe, or maybe the things that made you 'weird' then are fine now with different presentation or around different people, things can change and be different, recognise that those experiences now where you are developing confidence and feel honestly that you can impress are going to be as formative as those experiences in high school where you were rejected and felt alienated. And as much as there is alienation in society, shutting everyone out can only increase it; it's natural and okay to have boundaries, but learning to let people in and to seek people out is the only way to combat alienation.
There's no universal approach because people react differently to everything. This depends on you building knowledge of her, and vice versa, to converse with her. Some people hate talking about themselves, other people fucking love it and will super enjoy it if you ask them questions about themselves, some people it depends. Some people want to be asked their opinions on some things. I have no clue what the relationship between you is like, but I guess: try to move the relationship out of the classroom at a pace that is comfortable for you and her (even in groups initially). You have school and the school area and a host of mutual acquaintances and probably mutual interests to bond over, after all. The better you know her, the more honestly you can say you really like her, of course (not doubting that you are very interested in her).
So I can't give you any wonder advice sorry... but hopefully I gave you some things to think about, and some encouragement, because I think you're probably on the right track but need reinforcement and support, and maybe some advice sometimes on things where you're unsure or might be approaching it wrong (so post anything you're unsure about?).
I think there is something you are wrong about, for example: yes, you will probably be in the same program as her for three years, but 'screwing up' comes in many degrees and doesn't have to be permanent. You could do something really scary and transgress her boundaries in a way that makes her think you're a creep and then it's up to her whether she's cool... but that's probably not likely as long as you keep it in perspective, + same as for anyone (e.g. don't seriously offend and so on). If she rejects you, accept it and move forward; there's still the possibility of her being friends, and her getting to know you better. Maybe you'll move on while she's still getting to know you, and you'll both just end up friends and happy with that. Realistically: maybe she'll never be interested in you, but you'll end up with a super cool friend and get over the crush, or end up with someone else in your program or area. Doesn't mean you screwed up or lost, nothing permanent or irrevocable about it.
It helps if you're doing other things in your life though, and not constantly obsessing over her—not to just impress or even to be interesting to her, or even to give her space (although this is good because you have a year of uni on her and everything is gonna be new and exciting to her so sometimes, like, she's gonna wanna do other shit, let's be real), but also to enjoy your school experience and to give yourself perspective. For you, basically.
P.S. We used to have a relationship advice megathread, but it seems to have died in my absence.