To a friend's grandmother I am / I'd be more than happy to be of assistance is fine; to your own grandmother it would be too formal; I'd be / I am happy to help you is more appropriate. Does your grandmother speak English, though?
Of assistance is a postmodifying prepositional phrase; it is normally used with abstract qualities: a man of honor, a person of some repute. In I am happy to be of assistance, it seems to me to notionally modify I-- the whole nonfinite clause to be of assistance being a sentence complement.
I am in total agreement with you on this. When saying or addressing someone, with the phrase "more than happy to assist you", what you are saying is that you are qualifying the degree of happiness you are enjoying in helping someone. You can be "beyond" just normal happiness.
What would be a good/polite replacement for the phrase "I'd/'ll be happy to be of assistance"? The reason I am asking is not I am not happy and do want to attach any emotion or at the least not reveal emotion to the act of assistance.