Best approach for extracting bits from a certain kind of pseudorandom generator?

[grumpy]

Internal state aside, it absolutely IS a problem if a generator starts spitting out a huge run of zeroes - that could easily ruin a simulation.

That is a problem with design of the simulation, or expectations of it, not the random number generator.

Real distributions of (almost) anything do permit a non-zero chance of unlikely sequences. If you're doing something to prevent particular sequences (such as runs) then you're introducing a bias - regardless of the quality of the random number generator.

The problem here is nothing more than the fact that the larger state space is "encroaching" on the smaller range of numbers that we're pulling out, so we really just need a way to "shrink" the state space somehow. Neither lopping off zeroes nor using the proposed modulus/divide method seem to be the right solution though, so what?

As Mario said, it is easier to shrink the state space that the random number generator works with than to use a larger state space and then adjust the output to a smaller range (by lopping off zeros, modulo/divide, etc). The two are distinct, but you are treating one as the other. They are not interchangeable either (usually).