There are several different ways of producing subordinate clauses:
- with that ... that he becomes/should become a baseball player
- with a marked infinitive ... [him] to become a baseball player
- with an unmarked infinitive ... become a baseball player
- with for + a marked infinitive ... for him to become a baseball player
- with a gerund ... becoming a baseball player
Each lexical verb 'licenses' (permits) some of these, but not all. Want, for instance permits:
- He wants to become a baseball player ... with an implicit subject = the subject of the main clause
- I want him to become a baseball player ... AND
- I want for him to become a baseball player ... with a different subject
But you may not say
- ✲I want that he should become a baseball player.
- ✲He wants become a baseball player.
- ✲He wants becoming a baseball player ... and so forth.
And when you move the pieces around to create what grammarians call cleft constructions, like your What sentences, you have to follow the requirements of the verb, as in What he wants ... is to become.
Do is a little tricky, because it's not ordinarily a main verb; but with other verbs it acts like a modal, taking an unmarked infinitive: Yes, the paper shredder does cut the paper into small pieces. That's why, in your example, you may use either to become to agree with wants or unmarked become to agree with do. In the other construction, though, there is no such ambivalence, and you want the unmarked infinitive:
- What the paper shredder does is tear the paper into small pieces.
✲ marks an utterance as unacceptable