Do the -ing and to-infinitive "verbs" that follow catenative verbs always take the grammatical function of "noun"?

I'm wondering whether or not the verb form that follows a catenative verb has the grammatical function of a noun or of a verb, and whether or not it depends on the first catenative verb.

"I like to run" "I like running"

the -ing and the to infinitive can easily be switched in this sentence, and it seems to be implying "I like [the act of[running]]", so both the infinitive and the gerund are nouns.

"I like to run marathons"/"I like running marathons"

These seem like they might still be nouns but need "verb powers" to take the object, and then these sentences really turn into "I like [the act of [running marathons]]"/ "I like [to be in the state of [running marathons]]".

Is this true for both the -ing (and so it would be a gerund) and the to-infinitive? Are they, sort of, initially verbs so that they can take the object, and then the verb + object becomes a noun phrase?

If this is true, I'm also wondering if the same rules apply for a catenative verb such as "start".

"I started to run in the mornings" "I started running marathons in 2008"

"start" is a catenative verb that isn't a cognitive/emotion verb, and specifically suggests action. So what is the grammatical function of the to-infinitive and -ing verb forms? are they verbs? are they nouns? It just seems incorrect to say that their grammatical function is a noun, but everything I've seen on the internet is pretty vague. I've also seen that to-infinitives are never verbs, but only from one source and it just doesn't seem right to me.

So any ideas on the true grammatical function of the -ing and the to-infinitive after a catenative verb? And is it universal across catenative verbs, or does it depend on the verb category?

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.