So if you get dealt 4 to a Royal, pressing the DRAW button 3 seconds
later will have a different result than pressing it 3 minutes later?
No, it won't. The outcome is the same -- you have the same chance of turning over the last card to RF no matter when the identity of that card is determined. Given that the RNG is running through the cards a zillion times a second, and a millisecond's difference in hitting the deal button (or dropping the 5th coin) makes the determination, it's just not something that you can reasonably control. Even on the old old old machines, the RNG is still going a heck of a lot faster than you are.
The "psychological" thing here is even less appropriate. You don't, and can't, know what the last card is, no matter when you press the button. Therefore, it can't possibly affect you, even if it's indeterminate until the draw button is pressed -- because the next card out will always be the next card out, period. It can't affect your play, and it's not like you can count to 10 to make sure you hit the King of Hearts.
Wondering if it makes a difference *when* you push the button is like pondering whether God can make a stone so heavy that He cannot lift it. You don't know, you can't know, and it doesn't matter, anyway.