To efficiently encode XOR-constraint you will necessarily need extra variables (as you proposed) because we know that no bounded depth and polynomial size Boolean circuit can encode parity ("parity is not in AC0").
The problem with this kind of constraints is that they are not local and really disturb CDCL based SAT solvers because you cannot remove the constraint unless you have fixed all its variables. Now, cryptominisat has a specific focus on this kind of constraints and implements ad-hoc routines to exploit this routine. cryptominisat is hence focused toward applications where this kind of constraint naturally appears, be it circuit synthesis or cryptography.
Now, CDCL solver is another beast. Gaussian elimination / Grobner basis are not implemented in these solvers simply because it is "too costly". One way of understanding why is just to reframe what one call SAT solver. This is not really a tool to solve SAT (hell, they can't even deal with the pigeon hole principle without preprocessing), the purpose of a SAT solver is to quickly solve instances coming from natural encoding of constraint systems. In this framework, unit propagation and clause learning are basically extremely efficient because they uncover hidden structure in the way, us human, encode constraint problems. Hence (CDCL) SAT solvers are highly efficient in many applications. But it is easy to kill one with a few variables by simply going out of this framework. For example, they are killers in solving packages dependencies because the structure of these problems matches this approach, CDCL solvers are then just an extremely refined way of bruteforcing your way through the search space.
Now, if your application heavily contains XOR-constraints, then it is likely a (CDCL) SAT solver is not the right approach. For circuit synthesis, some people use BDD based approach (where parity can be efficiently encoded) but sometimes you will simply have to develop your dedicated solver or use other solvers for other NP-complete problem that will have a different area of specialization (as already observed in many comments).