Semantically, there is no real difference between the two constructions, and they can be used interchangeably. I agree with you that "should" has become the far more common construction.
Even back to the Old English (and earlier Proto-Germanic), the words carry a nearly identical sense of obligation. As such, I'd be hard pressed to articulate even a minor connotative difference between the two forms in modern usage.
"be supposed to do something", is required to do something because of one may find himself in a situation, or put himself in a situation: agreement, promise, and lack of authority if the usage in negative form: "I am not supposed to remove these books from the reference room". It is an idiomatic expression not a modal.
Modal verbs are must, shall, will, should, would, can, could, may, and might.
must, and have to are a lot stronger, the strongest is 'must'.
and there is 'should' in between have to and supposed to do.
You must obey the law.
shall expresses strong assertion, and promise in a positive form;
in the negative, a command, a divine order; thou shan't steal!
In the interrogative shall expresses an offer, " Shall I pick you up in the airport?"
ought to, comes close to should, as it indicates duty, and correctness.
Your usage is by the book.
Best Answer
"Should to" is never correct.