You could use that argument about any of the random numbers that your rng returns. "Hey, 0.023 is infinitesimally unlikely to occur, so let's exclude that as well".
Generally when I use a real-valued rng, the numbers generated are meant to be 'representative' of what I should get from the distribution. And when I get e.g. 0.023, it kind-of means "0.023 and/or numbers near to 0.023". If I excluded 0.023 from the possible results of the rng, then I would have a 'hole' in my distribution 'in the region of' 0.023.
Maybe what you want is to go from the possible rnds:
0.000, 0.001, 0.002, 0.003, ... , 0.999
to:
0.0005, 0.0015, 0.0025, ... , 0.9995
whereas with your suggestion, you are under-representing the boundary numbers near to 0 and/or near to 1. Generally not a problem I guess, but what you are doing is, imo, 'wrong in theory', even though your high-precision floats will probably cover it up ok in almost all cases.
I agree with you about 0 and 1 causing problems when they are fed into other functions - e.g. generating Gaussian rngs by using InverseCumulativeGaussian(0.0) - a bug I wasted some time hunting down in my company's rng library. My view was that the developer who wrote the library did not understand the maths of what he was doing, rather than that the 0to1 random number generator was at fault.
If you'll pardon me pulling out my soapbox for a moment, this kind of bug would be less common if the "programmers don't need math" attitude was not a successful meme in our culture. The notion of "between zero and one" may seem straghtforward and obvious, but is actually ambiguous. Perhaps, given the number of distinct floating point values that can be represented in the interval (0,1), it would hardly seem to matter what interpretation of "between zero and one" is used. But in mathematics you will never see such terminology used without a qualifying statement that clarifies* the meaning.
*E.g. "between 0 and 1, inclusive", "on the closed interval bounded by 0 and 1", "[0,1]", etc.