Why not do these checks on the number of clauses in 3-SAT?
There's no real reason not to do your checks given that they are computationally tractable, but there's very little point in doing them either.  Your checks make determining the satisfiability of easy random 3-SAT instances easier.  SAT solver authors are interested in algorithms that make hard SAT instances easier.

For random 3-SAT, the hardest instances are known to be near the phase transition point where instances on one side are almost always satisfiable and instances on the other are almost always unsatisfiable.  That transition point has been observed empirically to be when the ratio of the number of clauses and the number of variables is around 4.26.  Your checks would never be helpful with such instances.