No. The inputs are labour and capital, so that's already taken care of.

One answer might be "TFP reflects technical knowledge; the more knowledge there is, the easier it is to generate new knowledge". But you'd have to get into growth macro for the details. Indeed, Romer is the person.

It's not always assumed to be exponential. It was assumed exponential by Paul Romer because long-term economic growth is pretty exponential. Much subsequent work has followed that model, but not all of it.