If you generate a random number with equal distribution in a gigantic range, 0 is just as likely to happen as any other number in the range, right?

That's the meaning of "equal distribution".

In a working RNG, every number has equal probability.

But there's a probability <1 that the RNG is working as it should and a probability >0 that the RNG erroneously generates numbers that aren't equally distributed.

And because the range of possible numbers is so large and the probability of a mistake is not that tiny, a small result is more likely to be an error than a random result that happens to be small.

If I write a program to flip a coin and it generates heads 30 times in a row, is it more likely that this is just a random result or that I made a mistake in my programming?

One probability is 2^30, and the probability that someone makes an error resulting in this pattern is probably higher than 1/2^30.