Just want to give you a little background on how random numbers work. Actually, they are pseudo-random and there are arguments that there can be no true randomness in a closed system, but that's really a philosophical argument.
The "player action" is used to generate a seed for the random number. I'm not sure if you know anything about how random numbers are generated using computers, but it's pretty standard to use some uncontrolled, mostly random source to feed the algorithm that will produce the random number. One limitation is that if you input the exact same number into the random generator twice, it will yield the same results. For basic randomness, most operating systems use the current microsecond, which is generally random enough for most basic purposes. For more bulletproof randomness, then there are systems that use radiation output from slightly radioactive elements to generate the seed or input from some uncontrolled (by the program) source, such as the party poker function that uses the amount of time it takes to click on an action or somesuch. Since we're human and we can't do one thing *exactly* the same twice, then this produces a result random enough for the purposes at hand. In the end, the number is about as random as it can get.
However, there are limitations in the underlying computer system storage of large numbers that have been cracked. In the late 90's, there was an online site that a group developed an actual turn/river card predicting utility that worked because the underlying RNG was faulty. This specific RNG only had about 2 billion unique possibilities of orders of a deck, where there's actually 52! possible orders of a 52 card deck. 52! (52 factorial) is a number with over 26 zeros on the end compared to the 9 zeros at the end of a billion. See the huge difference in numbers? Since this particular site generated the deck's order once, you could succesfully figure out the order of the deck by comparing your holecards and the flop. By partypoker continuously shuffling the deck, so to speak, then the order is not set but each remaining card has an equal chance of coming into play. That's the reson that a set deck is not generated at the beginning of the hand.
Partypoker actually gives out all their RNG information on their website, if you're willing to believe what they say. Before I ever looked into playing online poker, I looked into the randomness of the shuffle because I wanted to be certian and confident in the shuffle myself. it'll look like a bunch of mumbo-jumbo if you don't have any computer backround, but here's the link.
http://www.partypoker.com/about_us/rng.html
It should also be pointed out that a live shuffled deck is not truly random, but again it's random enough for the purpose of fairly distibuting the cards.
If a casino wanted to shuffle the deck the whole time, as long as only the burns but not the muck were included back into the deck then it would not be a problem because as you said, the odds of one specific unseen card falling would not change in the slightest. You would still have the same expectation and equity, much similar to how "running it twice" works.
