Ending a sentence with "don't want him to"

John is not doing his job as I instructed him to.

John does things that his parents don't want him to do.

Greetings.
Why is it that in the first case you can end the sentence without a do, and in the second case, you have to have a do?
The explanation I heard is that if a sentence contains a relative pronoun (as in John does things that his parents don't want him to do) and the verb involved has the form of "to do something" as opposed to "to something," you cannot omit do.

For example,
John does things that his parents don't want him to do<-- want to do something, not want to something, so need a do
This is what it boils down to. <-- boils down to something, not boils down to do something, so don't need a do

Is the explanation correct? Could you give a more clear explanation?
Your help would be appreciated.

Thank you