Guess what? I would like to counter that argument with a reference from another novel of the time! In Vanity Fair Amelia goes to a very similar school and learns how to sing. She also is no prodigy. When we see the school ads from the period in places like JA Regency World magazine, they always list singing in the things they will teach. I mean how hard is singing? I can't say Harriet wouldn't be too shy to play or sing but I think it likely she would have been taught.

Ah, to give a JA reference. Elizabeth Bennet sings and plays without the benefit of a governness, school or master. She seems like someone who would have taught herself well anyway.
Generally a "song" inferred singing whereas today we don't think of it like that. A piano piece that wasn't meant to have singing might be a sonata or etude.