Sounds like a neat idea that complexity theorists should have come up with in the 80's. And we did, where by "we" I mean Larry Stockmeyer in his 1985 SICOMP paper On Approximation Algorithms for #P. Stockmeyer uses random hash functions from Sipser but it is essentially the same algorithm and analysis as Gomes et. al.
Stockmeyer wrote a purely theoretical paper. Since then we have better algorithms and much faster computers and one can now solve satisfiability on non-trivial input lengths. Gomes et. al. write the paper as a practical technique to approximate the number of solutions and even implement their algorithm and contrast with other programs that exactly count the number of solutions.
Gomes et. al. mention Toda's paper on the hardness of exact counting but don't cite Stockmeyer or any other paper in the vast theory literature on approximate counting.