Is it normal that i am great at teaching but that i don't want to be a teacher?

If you don't want to do it, don't do it. But if you do want to do it... there are things you can do to try and mitigate your worries.

Firstly, your attitude of not giving up on people even if they look like failing is a good thing. Good teachers shouldn't give up on students, otherwise the students have no motivation. Not wanting to give up motivates you, it motivates students too. Students want to work for teachers who believe in them.

Secondly, have a back-up plan. If you're good at tutoring you're probably already intelligent enough to have more than one career option. Having a back-up plan means that if teaching becomes depressing for you you have other jobs you can turn your talents to. A lot of the (good) teachers who taught me only did teaching for five years before moving onto something else.
You don't need to be stuck in a teaching job for life. These days it's common for people to have many careers in their lifetime, and experience teaching is something a lot of employers like in jobs completely unrelated to teaching. It tells them you're confident, able to think quickly, able to plan long-term and in-depth, able to present, and good with people.
Another option is to regularly move to different schools. I don't know how it works where you live, but a lot of the teachers who taught me had worked in five or six different schools before. If you dislike the atmosphere in one school, it isn't too difficult to get rehired at another next year. It's a revolving door. I had one teacher who never taught at the same school for longer than one year and she'd been teaching for almost ten years. It wasn't that she kept getting fired. She just liked to move around a lot and could because really good teachers are in demand.

As for uninterested students... I suppose you're bound to have some. You can convert some of them, but some just don't want to be engaged. All you can do is work your hardest for them and hope they respond. Most students respond to a teacher who cares about the subject and cares about their future.

I'd like to be a teacher too, so I've thought about this a lot :P

You're great at teaching and also great at being modest

I been told I should be a teacher but I really not interested in that type of career. Apparently people like how I explain things. Simply having the talent for something does not mean you care for it. I would rather be doing something more direct.

This is a problem. Teachers in school suck and people who can teach dont teach.

Guess it depends on what you want to teach. From there, you can choose your audience. It sounds like the only option you are entertaining are public schools (or private) that are the basic classes (math, science, history, english, etc). But if you went into something more specific, you're bound to get students who want to be there more often then not. And yes, it's pretty normal. Another example would be the artist or musician who does not want to play professionally at the cost of their love for it.

The way I see it, if you like to teach but don't want the hassles of being a babysitter in a public school, you could try teaching in a college or a private school. Not being at all familiar with that profession, perhaps you must "intern" in the public school system. If that is so, then the system is losing out on some very good teachers like yourself, who have no interest in being paid poorly to babysit those disinterested and disrespectful students. What a shame, but it's just another thing in the US that is broken.