Would a Latin have recognised that fari, fama, and fabulis were related witds?

I think it depended on the speaker - writers such as Cicero and Ovid thought about their language - and on the transparence of the word: 'fābula' is recognisable as 'fā-' plus '-bula', a suffix also found in other words, but 'fāma' may have been a bit more difficult.