SAT is NP-complete, and both gaussian elimination and 2SAT are polynomial-time, so this would suggest either you're mistaken or there is some hidden catch here (like the size of one or the other being exponential-sized).
There is no catch - I even describe the reduction in another comment below. You can convert a 3SAT clause to a combination of XORSAT and 2SAT clauses. I am not mistaken, either, I used this reduction many times on practical problems, so I know it works. I encourage you to try it.
Unfortunately, putting the algorithms for XORSAT and 2SAT together is not trivial at all, they are quite different (but Grobner bases over GF(2) seem very promising in that).
But I agree that the fact that both XORSAT and 2SAT have polynomial algorithms is quite a strong indicator that full SAT has a one too. :-) (On the other hand, there is IMHO only very little actual evidence for P!=NP.)
Unfortunately, putting the algorithms for XORSAT and 2SAT together is not trivial at all, they are quite different (but Grobner bases over GF(2) seem very promising in that).
But I agree that the fact that both XORSAT and 2SAT have polynomial algorithms is quite a strong indicator that full SAT has a one too. :-) (On the other hand, there is IMHO only very little actual evidence for P!=NP.)