But if there is no obviously desirable move, and instead a wide array of potential equally good moves, then picking the same as the computer picked gives a high probability of cheating. That being said, I find it unlikely that that's the case here.

I believe you're starting from a false premise.

In case of the first moves, there's definitely not a wide array of potentially equally good moves. There's a very well known set which is studied by every serious chess player.