Wait for - Why?

The wait pattern uses a FOR...TO... clause. The subject comes after FOR. The verb comes after TO.

I waited for him to finish. = I waited for this: He finishes.
After the verb wait, the clause he finishes becomes for him to finish.

These can also show the purpose of the action. That is, the clause often answers the question Why?

I stepped aside for them to go by. = I stepped aside for this: They go by.
Why did I step aside? So that they could go by. = For them to go by.

The teacher gestured for the boys and girls to come in.
= The teacher gestured for this: The boys and girls come in.
Why did he gesture? So that the boys and girls would come in. = For the boys and girls to come in.

The FOR...TO... clause is much more common after adjectives:

He was eager for us to meet his brother.
It is important for the students to study hard.
It was not possible for people to buy tickets before noon.
__________

In the buy pattern there is no clause transformed into a FOR...TO... structure. FOR simply indicates the person who benefits from the action. In a way, it shows the purpose of the action by showing who is going to benefit from the action. A FOR phrase can be used after any action when you wish to mention the person who receives some benefit or advantage or favor because of the action.

I bought it for her. She got it for me. She wrote down the address for her sister. We cooked dinner for them. Joseph painted the kitchen for his wife. Susan saved a piece of cake for me. My brother kept the money in a safe place for us. My cousin will watch the children for me. We should hide the money for them. I cleaned the table for her. I can do a favor for her mother.

Other patterns, such as the pattern with give, simply indicate the direction of the action with a TO phrase. Here TO is followed by a noun, not a verb as shown for the FOR...TO... clauses (above). There is no sense of doing a favor for someone, as in the FOR phrases (above).

I gave the shirt to him. I showed the answer to Mary. He sent the money to the vendor. Gary mentioned the book to a friend. Sue made a promise to her mother.