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.
According to Cobuild (Collins) (Chapter 1:7), the verb + -ing form catenation has three structures (not confusing the non-catenative strings such as Under the trees Bill strolled, looking at the flower beds):
[examples partly adapted]
[1] Verbs in phase (ie a two-verb structure where essentially one two-part concept is expressed):
The sea came rushing in.
He started / kept / stopped crying.
She avoided looking at him.
I won't bother going.
Have you tried asking?
I'm going shopping.
(These mostly invite echo questions such as 'What did he start doing?' 'What won't you bother doing?' With the 'going fishing' type, the echo question would be 'You're going ...?)
[2] Verb with object (ie with what is often termed a gerund)
I like being alone.
Have you considered applying?
She recommended staying.
He didn't remember leaving.
This involves stripping down the engine.
(These mostly invite echo questions such as 'What have you considered?' 'What does this involve?')
[3] Verb with adjunct ( depictive or resultative)
The soldiers died fighting.
Their boat finished up pointing the wrong way.
(These mostly invite echo questions such as 'How did the soldiers die?' (ie What was the manner of their death) 'How did the boat end up?')
I'd argue that these usages are [1] verbal, [2] verbal-nounal, and [3] adverbial or adjectival.
Best Answer
Another way to think about it is as a subject complement.
The man (Subject) tried (verb) to run (infinitive) faster (Adverb).
The complement refers to the subject.
An example of an object complement would be:
The people chose the candidate who was tallest.
In this case, who was tallest? The object - the candidate.
In the first example, the subject is running.
I don't know if this helps but when I am stumped, I often try to substitute concepts or in some cases pronouns or tenses to test various grammatical problems or to help my comprehension.