Is 'that' required? Does it make a difference?

Solution 1:

I want that he become a doctor. --> No problems.

I want he become a doctor. --> Ungrammatical; alternative could be:

I want him to become a doctor.

Usage:
Raymond Martin, Self-Concern, 1998, p.39

A person may want that he be the one to finish it so that he personally is responsible for its being finished and/or so that he gets credit appropriately for its being finished.

ibid., p.70

… he can show that C1 could not want that he individually survive or that both fission descendents survive, but at most that at least one of the fission descendants survive …

[EDIT]
There seems to be some objection to an ECM verb like want used in this structure, while the other verbs like insist present no problems.

I insist that he become a doctor.
I insist that you try some of this cake.
I insist that he come to our party.