I have a vague memory that Weil (or someone else) actually complained in print somewhere about people making conjectures unsupported by much evidence, and using Mordell's conjecture as an example, with words like "there is no evidence one way or another". If it is Weil, it is probably in his Collected Works, which I don't have handy to check. Obviously this was before Faltings's proof.
Most 'heuristic' explanations tend to suggest that failures of the Mordell conjecture should be rare, but don't really say why there shouldn't be any. For instance, one can give a pretty good heuristic that squares of integers don't exist (they have to be square modulo every prime, which means surviving infinitely many coin-tosses which are more or less independent).
A last remark concerning JSE's answer: I remember from a lecture of Faltings that he was saying that he and/or a student of his were working hard at trying to generalize Kim's method to all number fields; off-topically, the following is -- as far as memory allows -- one of the pithy things he said: "Motivic cohomology is a bit like communism; in principle, it's great, but you can't compute anything").