How can I take an interest in my students without being precieved as intrusive or weird?

Hi! I (24M) just finished my internship at the high school where I studied, and I loved it. To give you some context, I’m a Spaniard (so obviously all happened in Spain) and I’m studying to become an English teacher. Even though in Spain the process to get to the public system is quite complex to say the least, I believe it'll be worth it.

So my problem is that I wish I had shown more interest in my students to get to help them. There were some occasions where I could sense that something was off, but never acted on it. The thing is that I always thought I might be bothering them or that they would think I was being intrusive. Even though they were always open with me, most of the time I never went past generic conversations. There was this very energetic 13-year-old girl (except when it came to homework) who would occasionally ask me strange questions or invite me to dance (tiktok and all social media to this generation is just something else lol). Once she started eating a homework sheet, and previously she talked to me about her favorite liquors…

I never asked her personal questions out of fear, but now I honestly regret it. They're children after all, they all need a figure to look up to and I feel like they want to reach out to us in some way but they don't even know how or why.

There were more cases but... Anyway, I just believe that many of them need someone to guide them. I know I can’t help everyone (especially because most of them will reject any approach right away), but I don’t like pretending nothing is happening and just being there repeating how to conjugate an irregular verb lol (which is important too).

So… does anyone know how to approach these situations? How can I show that they could rely on me if they wanted to without seeming intrusive or weird? Is it really worthy to get involved in their lives?