To be fair, one specific password does tell you something about the entropy of the generator. If I can grab 1 MiB of data from your password generator, I can probably estimate quite well the entropy of the generator. 10-20 bytes? That tells you almost nothing (but still not nothing). That means there must be some way of updating an estimate of the entropy of the generator from every additional byte of output and the amount of confidence that you gain about that estimate grows (probably exponentially) with every new byte.

But your password is 10-20 bytes so you can say nothing about the generator.