Yeah. For the most part non-constructive proofs are only good for proving things about existing algorithms - their convergence properties, their correctness, their running time etc. Formal proof often isn't necessary in practice, though - "seems to work well enough" is ok in many domains.
Of course, this isn't to say that new mathematics based on the result and on the techniques used to prove it won't lead to new algorithms.
Of course, this isn't to say that new mathematics based on the result and on the techniques used to prove it won't lead to new algorithms.