The NIST contests are great, but I would expect that running such a contest for a hash or a cipher is easier than doing so for most protocols. It is possible to define exactly what a cryptographic hash must do in order to be considered a cryptographic hash. I think one would have a harder time making an analogous characterization of the solution space for HTTP security.

Whatever eventually replaces TLS, I doubt it will be something that could have emerged from a limited-duration contest.