It's not a stupid idea, it's a problem that smart folks have been banging their head against for a long time now. Take a look at the "TLS Snap Start" proposal to see the lengths to which one must be willing to go to avoid that round trip.
But some low-hanging fruit remains. Improvements to clients and servers that increase TLS session resumption rates would help too.
But some low-hanging fruit remains. Improvements to clients and servers that increase TLS session resumption rates would help too.