Is this possible to win on all computers and without inordinate amounts of luck? That is what I must ask.
I am sure there is some exact hardware + OS + browser combination in which it is possible, but my hunch would be the time runs faster for some people than others and the lag is worse for some than others, all of which adds up to less time to complete stages for some, to the point perhaps some of the later stages literally aren't possible on some hardware +OS + browser combinations? It's a problem with any tightly-timed games of which this definitely is one.
The game does lag up a bit, which really isn't acceptable for something this simple. You have one screen and one screen only. There also isn't that much in terms of animation going on (exception being when you flip a card). There's no reason this game should have any lag at all.
As for luck, obviously that's a factor in any memory-matching game where you can get lucky and sometimes correctly-match the first time you flip cards. That's not an issue. What may be an issue is if such luck is *required* in order to complete the later stages -- in other words if they can't be done simply with flawless memory itself (That is, no mistakes on cards you've already seen at least once).
So, can this game be done for all setups and using nothing but memory? I don't know. Very few people have done it and obviously those people have excellent memory, but I don't know if they also needed large amounts of luck and/or some specific hardware + OS + browser combination.