[FlFishOn]

"As long as you realize that your projected sample size is still painfully inadequate to make any sort of conclusions on the reliability of a random number generator, then knock yourself out."

This is simply wrong. Let's take a look at your coin analogy. We will wager on a coin flip. I'll be flipping the coin. No, you may not inspect it. You may bet heads only on any flip or pass. After 20 flips, 10H/10T you jump in and bet and lose. another 20 go by, 11H/9T and again you bet and lose. Repeat this 20 times, you're -20 bets. Still want to bet? No, you'd be a fool. In fact you likely quit betting after you lost 8 in a row. You drew your conclusion after 8 flips!

"You will not be proving anything either way and any conclusions you draw will be based more on your a priori arguments than any sort of true statistical analysis."

Bullsh+t. With proper analysis reasonable conclusions can be drawn from scant data. It's strictly a question of understanding the statistics of hypothesis testing.