The part you're quoting is the spec, not the implementation. It implicitly assumes a machine model where sampling a uniformly random real number in an interval

The part you're quoting is the spec, not the implementation. It implicitly assumes a machine model where sampling a uniformly random real number in an interval and correctly rounding it to a floating-point number are both supported operations. Of course this cannot be a real machine or even a Turing machine, but you can imagine and mathematically model a more powerful machine which can do these things. (Turing machines are imaginary anyways.) This unimplementable algorithm for an imaginary machine is still useful, because it nails down the probability distribution we expect the output samples to have. And because they're sampled from a finite value space and the probabilities of obtaining each possible output are rational numbers, we have a chance of finding an implementable algorithm for a real machine that produces the same output distribution. The rest of the article goes into detail on what they decided to actually do. It's similar to your proposal but has more edge-case handling and deals with different rounding modes.