This is another "get" pattern which is not commonly taught. It is a colloquial pattern, and this is perhaps the reason why it is not taught.
"I got to do it" has two, but overlapping, meanings:
1) I got the chance to do it./I was lucky enough to have the chance to do it.
2) I got the time to do it./I finally found time to do it.
As I said, these are overlapping meanings, since having the time to do something also often means we have a chance to do it.
This "I got to do it" pattern can be thought of as being a pattern in which "chance" and "time", or similar words are omitted.
Here are some example sentences with these patterns. As the reader will see, sentences (3) to (9) are more closely connected to the idea of chance, and sentences (10) to (16) are connected with the idea of time.
3) When do I get to be batter?
4) I got to play a small part in the movie.
5) But Dad! All the other kids get to stay up late on Saturday night.
6) During this cooking class, you will all get to bake a cake.
7) You get to do it the way you think best.
8) In this game, you only get to throw the ball once.
9) We always do things your way. When will I get to do things my way?
10) I finally got to clean out my room last week.
11) If everything goes well, I should be able to get to work in the garden this weekend.
12) I just cannot get to write the new computer program.
13) Don't worry. There is plenty of time. Everyone will get to have a go.
14) I just cannot get to do all the things I want to do.
15) You are always making me hurry with my work. I never get to do it properly.
16) I have been really been busy and have not had time to get to do it.
・be thought of as ～: ～と見なされる（＞think of A as B）