Right, but storing 52! distinct values isn’t something a computer is going to be able to do.
The whole premise of computers being unable to do shuffling as good as humans is ridiculous. 52! is huge but log(52!) is not.
If anything it's humans (unless specifically trained) who produce horribly biased shuffles. Whole chunks of sequences from the previous deal can show up, either directly or spaced 2x or 4x. Try ordering a deck first, then shuffle it, then open each card and see if you notice any patterns.
You’re only using one deck at a time; so you only need to generate 1 bit randomly 226 times — then use that deck.
There are strong monetary incentives to break all kinds of private keys but most of the time they hold pretty well.
(That does sound like a fun optimization challenge: "write a program to fairly shuffle the most decks of cards per second possible.")