Learn English – Is “ran after” a phrasal verb

The informal rule is a stylistic one. Keep the complement as close as possible.

That really helps me out.

Clearly this is not a lot of separation, and to phrase it "helps out me" would sound awkward and awful.

That really helps out the children who are starving every day in Africa.

To put "out" at the end would simply require the reader or listener to wait too long to parse your verb as a phrasal verb.

To sum it all up: it's a judgment call.

To sum up everything I have stated in this response: it's still a judgment call.

Taking your example: "[This is the] man who the authorities refused to admit existed," you have correctly identified that you want to look at the variation "The authorities refused to admit he existed." In that variation "he" is a subject of a clause. "Whom" is used for as a pronoun for the object of the dependent clause, not when the clause itself is the object of the independent clause. So, while "he existed" is the object of the variation, "he" is the subject, and that's what matters.

To take another example:

This is the man whom the doctors refused to admit.

is correct because "the man" is the object of "to admit." In your example, they were not refusing to admit the man, the were refusing to admit a statement about the man.