Firstly, I should say that I am a learner as you, but I want to share my thoughts. I wish you and others help me to improve my answer by your comments.
I start with the definition of advocate:
to publicly support or suggest an idea, development, or way of doing something; [Transitive Verb ]
Based on my google searches, you can use advocate mainly in five ways.
Thus, if you ask whether or not it is grammatical I say yes, but if you ask which pattern is more natural among the others, I am not sure.
In terms of your second question, gerunds can act as nouns and they can be used after verbs, but the question is where.
They can follow certain verbs having certain prepositions (object to, insist on, ...):
He confessed to stealing the money (to is preposition)
They can be used after objects with perception verbs (watch, see, hear, feel, ...):
I saw them running on the bridge
They can be used after verbs having liking or hating connotation ( detest, love, ... )
I like Tom winning the match
I hate Tom's winning the match
They can be placed immediately after the verb, acting as an object (similar verbs, enjoy, finish, consider, imagine, ...)
You should avoid driving when you are drunk
However, as @TRomano mentioned, a transitive verb needs an object and a gerund can be used as the object since they act as nouns. There is no specific list, but I agree that we can use several common patterns and verbs to convey our intents to the native naturally.