If they don't know they're watching someone with an accuracy of 60% over hundreds of shots and instead have to judge on nothing but 10 consecutive hits, it's actually perfectly rational to overestimate his ability, though it should of course have a low confidence and be constrained to the known possible range.

I'm not sure that they are:

Yes the most likely time for him to score, all else being equal, is just after he's scored. This is really just looking at the cumulative probability of something not happening - the chance of not scoring on consecutive attempts goes down just as the chance of scoring on consecutive events goes down - (with the remainder constituting the other potential arrangements of observations.) And so observations for given events have a certain tendency, for a certain level of probability, to occur in clusters - which influences how much confidence you can place in something given the data.

However, that does not make it likely that that'll keep happening for large runs, it's more likely that a yes will fall after a yes than that it'll happen elsewhere, but more likely is not the same as very likely. And the more observations you make the less likely it is that you're just looking at a cluster that'll occur naturally in a given number of cases.

By way of a, perhaps simpler, example: if you flip a coin and it comes up heads six times out of ten, that might just be a fluke. If you flick a coin and it comes up heads six thousand times out of ten thousand, then you can be fairly confident that it's crooked.