In the phrase be busy doing something, doing is a verb in the -ing form. It tells us what the person is busy with. For example: • She is busy cooking dinner. (cooking dinner tells us what she is doing.) “Grammatically speaking, what function does "doing" have in this phrase?” Tip: When asking about grammar, phrases like what function does… have or what role does… play sound more natural than what does… work as.

I am happy to help. Please listen to my mini PodCasts 😉

I would say as a gerund I am busy swimming I am busy doing the dishes doing - some action or activity = verb in the ing form = gerund :)