Being resistant to injury due to flexibility and balance would enhance performance.
No it wouldn't - it would just mean you 'might' be more resistant to injury, nothing more. It would not give you a performance benefit in the sense that it would not make you faster. Being resistant to injury is not going to win you a 2-up sprint against another guy who happens to have done more training.
It's a bit like saying wearing a helmet might improve performance, as it might prevent a head injury if you crash - which it probably would, but that's not the point - you don't wear a helmet as a performance enhancing tool, unless it's a pointy hat for TTs. And at that point, it pretty much ceases to function as a safety device anyway.
Anecdotally, I have never had a leg injury through cycling, and I have never been to the gym either - so any preventative measures in the gym that I might have undertaken would have been a total waste of time. If the OP wants to talk about preventative exercise, then that's a whole new debate. But until then let's stick to the issue.