StoreGate: disable implicit string to HandleKey conversion

Disable the implicit std::string to HandleKey conversion as this can lead to surprising failure modes as the implicitly created key will not have been initialized correctly (see e.g. !77957 (comment 9103142)). This also requires the addition of a push_back(const std::string&) overload for the array use-case as we can no longer rely on the implicit conversion.

Also add a few missing assignment operators.

Doing this in 24.0 to avoid surprises when code gets merged into main.

cc @ssnyder