But the probability that a true random generator will output 0000000000 should be the same as the probability that it will output 0010111101, because all sequences of equal length are equally likely.

With a fair random generator:

p(0000000000) = 1/2^10
p(1000000000) = 1/2^10
p(0100000000) = 1/2^10
p(0010000000) = 1/2^10

The numbers produced are independent of each other and for our purposes we don't care about the order. The relevant thing is how likely it is is to produce a given total number of zeroes or ones.

p(just one 1) = 10/2^10; A whole heap more likely!

So the chance that the generator is fair is rather slim. You can calculate just how slim by simply applying bayes rule (and doing some integration).

On a related note if you role two six sided dice you are just as likely to get two sixes as you are to get a three and a five. But if you are playing Settlers of Catan and put all your settlements next to the twelve instead of the eight then you are probably going to lose.