(edited because what was intended as succinctness read like approbation)
SoB is always looking for ways to speed up the search. Some large improvements in sieving speed happened in part because of discussions on these boards. However, the existing speed improvements have already taken us beyond where this particular idea would be helpful. There are two major reasons why we have already outrun this idea:
Firstly, sieving is about checking whether the numbers are divisible by "small" primes, and that process is already far past the stage where anybody has simple rules.
Secondly, modular arithmetic gives us a way to determine the remainder after division that is much faster than creating the whole number. When the remainder is zero, the number is a divisor.
So no, these rules have additional insight for SoB.