There are various reasons you might use a random number generator. If you're doing a 1-D Monte Carlo integral you care about "uniformity" but don't give a hoot about long-range correlations. If you're seeding cryptographic keys you care only weakly about uniformity, but you want a long period.
What's the point of the random numbers for Mr. Powell? We care that his receiver can't exploit a flaw in the number-generators to "guess" what number Mr. Powell has been shown. For example: if the receiver knew that the numbers were being picked "randomly" by a human, he'd do well by guessing 7 and 3 often, 1 and 5 rarely. If the reciever a) knew that the numbers would be generated by a certain implementation of /dev/random, b) knew that the random-number seed would be the time of day in seconds, c) could guess within 1 minute what time the seed was drawn, and d) had a confederate or concealed computer feed him the 60 possible sequences, he'd have a 1 in 60 chance of winning.
Seriously, though, none of the random-vs-pseudorandom arguments above are at all relevant to this aspect of randomness. Since the guesser gets no feedback, we don't need to worry about correlations. Since we're only looking for 1e-6 odds, we don't care much about period. To avoid the guess-the-seed problem, at worst, we need only make sure that there are more than 1e6 bits of uncertainty in the seed---not good enough for cryptography by a long shot, but fine for us and easily achieved (for example: seed the RNG with a ten-digit hex number pulled from dice rolls.) We care about uniformity at the level of a few percent---as long as the generator doesn't land on one number twice as often as another, we're fine.
In other words: dice are fine. Cards are fine. Pseudorandom numbers are fine. Tables are fine, again, as long as the receiver can't guess (with 1e6 odds) where we're going to start in the table.