Did you mean the girl doesn't have to go to school on any Saturday or just one?
I do not have to work on Saturday (a specific one) does not mean I do not have to work on Saturdays ever.

interesting
It does not have both meanings to me.
She does not ever have to go to school on Saturday. = She does not have to go to school on Saturdays. (plural)
She doesn't have to go to school this Saturday. = She does not have to go to school on Saturday. (singular)

very interesting question

I can only speak for myself. It is not my inclination to say that, and as I actually thought about it, I am not accustomed to hearing it either.
As an English teacher, I want to fix it by changing the tense or adding an s.
The tense, not any other aspect of the verb, indicates routine or habit, and habit means more than once, so the plural seems correct unless a modifier is used.

I go to church on Sundays. I go to church every Sunday.

every Sunday = more than one Sunday

I will send a message to you. Yes, people should not always write as they speak. What is commonly said or accepted is not always correct. Poetry (like lyrics) does not need to conform to any rules.