Without commenting at all on the practicalities of attacking it, it's reasonable to expect a higher level out of security from GPG than a standard TLS connectio

Without commenting at all on the practicalities of attacking it, it's reasonable to expect a higher level out of security from GPG than a standard TLS connection. Your web server needs to key and manage many connections/sec, and any delays feed back into my web developers further arguing against the use of TLS. You can generally expect no one will be trying to break your current TLS sessions ten years from now. If that doesn't apply to your situation, you probably have bigger problems, such as the broken CA model. Both of these scenarios may be different with GPG.