You'd think that the handful of people who devote a significant fraction of their time to writing SAT solver would be aware of such simple tricks and there might be other reasons why your idea isn't being applied.
I would think so too, but I couldn't find this particular trick anywhere.
Anyway, I believe it's similar to LP - we still use simplex method in practice, despite the fact that LP has a polynomial algorithm.
What likely happens in SAT is even though there actually is (as I believe) a nice polynomial algorithm on the order of O(n^4) or so (which involves repeated solving of linear systems), the CDCL still wins in practical problems simply because in most cases, the upfront cost of solving the equations is higher than trying to find a first solution.
However, not all hope is lost. You could actually presolve a more general problem, and then just rely on propagation to deal with a more specific instance. For instance, for integer factorization, you could presolve a general system for integers of certain bit size, and then if you wanted to factorize a specific integer, you would just plugin constants into the presolved instance. That would eliminate most of the upfront cost and bring it on par with CDCL.
Unfortunately, DIMACS is not all that good format to represent SAT, because it is not composable like this - it doesn't allow for representation of presolved instances. But composability might be a win for a different types of algorithms, which do presolving.
I don't really know the exact algorithm yet, but I have a strong feeling it is possible based on what I see. There are like 3 major steps in the strategy, first one is 2XSAT reduction, second is a proof of refutation completeness (in particular, a form of deduction theorem) of a certain logic that models 2XSAT class, and the last one is polynomial-sized representation of all true statements in that logic (which uses n sets of linear equations, and from that follows the complexity of O(n^4)).
The 2XSAT reduction is 100% correct (I have been using it to solve small integer factoring problems), the deduction theorem proof still has a small flaw which I think is fixable, and the third part is an ongoing research which is trucking along. I will get there eventually, but if more people would be looking at this approach, then we (humanity) could arrive there faster, that's all.
