Randomness isn't independent of process.

That's right. And more, randomness is pretty much independent of the result, with small samples.

Say, I play on the lottery. I pick 1,2,3,4,5,6. I win. Will you say the result wasn't random? And 1,2,3,4,5,6 is just as probable as any other winning sequence!

If you see a table where 200 coins lie heads up, you can suspect the process wasn't random, and the probability that you are right is about like 2^200 to 1.

Toss 200 coins and if not a single tails show up, the result is still random, just extremely unlikely. Toss 6 dice and get "1,2,3,4,5,6" and although the result seems suspicious, it's definitely random, because the process was random. You could say it wasn't random, if in 2000 tosses you got this result in 1/6 cases and not in 1/6^6. And still you'll be speaking with some degree of probability (very big), not with certainity.

Show me 30 samples from your "unsatisfactory random number generator" and I'll tell you you probably had bad luck while generating them. Show 200 and we can start talking about the generator quality.

(I used to play RPG games. There's a lot of dice tossing involved, and although I have no doubt about randomness of the process, I've seen amazing wonders when rules of probability in short term were out to lunch. Still, we just laughed and nobody requested "replacing the fake dice".)