Does she want you to cold approach her? Well, it does not matter what she secretly wants, what matters is what you want.
You don't invite someone to a party because you think they are wanting to be invited out, you invite them because you want to invite them and either they accept or decline, same goes for cold approaching. You approach because you want to, not because you think they are waiting for it.
The reason you do cold approach is not because it's normalized in your area or because somehow have the superpower of reading minds, you cold approach because you want to be the type of man who does what he wants, including approaching a woman that he likes, even if the rest of society finds it weird.
See i believe most men problems and frustrations come because they never actually do what they want to do, they always do either what society tells them they should do, or what a particular woman tells them they should do. But what about what he naturally wants to do when no one tells him simply because his heart calls for it?
They never do that out of fear of being ostracized and negative reactions, and this creates weak men who are always reactive instead of proactive.
I realized that it's irrelevant if something is weird or not, because it's not your duty to be normal, or avoid being weird. What's relevant is doing what you want simply because you give yourself the authority to do whatever you want to.
It's about empowerment and authenticity, it's about being the type of man who follows his own rules, not other people's rules. A guy who wants to have the integrity to honors his desires and does not let society, men or women, tell him what he has to do.
A guy who does not conform to arbitrary social norms that are not his obligations to follow. A guy who has the full conviction that what he does is valid, not because others approve it, but because he approves it unilaterally simply because he says so and that's good enough.
Even if the girl herself questions why you do would do it, you don't go defensive, you don't apologize for doing what you want, you don't justify yourself, you don't stop doing it, you simply double down because that's what a man who knows what he wants and respects himself does.
Because you don't need her permission as no one has granted her the divine authority to dictate to you what you can or cannot do. She can only refuse what you offer, not tell you what to do.
So it's about being a man who leads his own life on his own terms, not a dude who always does what others expect from him or who lets society or a woman dictate to him what he has to do.
You might then recognize that it's not even about whether a woman likes you if you approach her, or ask her out or invite her to your place, it's about what type of man you want to be. The one who has the independence and willingness to go against the norm and do what he wants, or the guy who hates himself because he is always doing what everybody else wants at the expense of what his own heart wants to do in the moment.
That's why you do this, because the proactive mindset of acting because you want to rather than because someone told you, later translates to other areas of dating such us who is leading his own life in a relationship vs who is letting her lead his own life for him with the added consequence of her losing respect for him for his lack of independence.
If she rejects you, that's fine, that's her job. Respect her decision gracefully and move on. But it's not your job to reject yourself before she does.