As the number of clauses increases, so should the number of variables (randomly chosen). That is, the alpha that your result mentions can well fluctuate between over-and-underconstrained values even with a googol clauses.

For any particular 3-SAT instance, you can calculate α. If the random algorithm that generated it doesn't favor any particular clause types, then the distribution of outputs with c clauses and v variables should be uniform among all such 3-SAT instances, and therefore we can use α to figure out if the formula is (most likely) satisfiable.

Furthermore, unless you deliberately try to generate random instances with a particular value of α, I imagine under most models the value of α will be (with high probability) well on one side of the threshold, and so the probability that your formula is satisfiable will be either very nearly 1 or very nearly 0, depending on the random model.