While trimming fat from excessively complex web pages would be nice, HTTP over TLS over TCP has some very clear fundamental inefficiencies that need to be addressed sooner or later.

The biggest one is that there is just no reason to negotiate multiple encrypted sessions between the same client and server just to achieve multiple parallel reliable streams of data. But since TLS is running over TCP, and a TCP connection has a single stream of data (per direction) you are forced to negotiate multiple TLS sessions if you want multiple parallel streams, which needlessly uses server resources and needlessly uses network resources.

Additionally, because TCP and TLS are separate protocols, TCP+TLS connection negotiation is needlessly chatty: the TLS negotiation packets could very well serve as SYN/SYN-ACK/ACK on their own, but the design of TCP stacks prevents this.

I believe it would be theoretically possible to simply set the SYN flags on the Client Hello and Server Hello TLS packets to achieve the TCP handshake at the same time as the TLS handshake, but I don't think any common TCP/IP stack implementation would actually allow that (you could probably do it with TCP Fast Open for the Client Hello, but I don't think there is any way to send the Server Hello payload in a a SYN-ACK packet with the default stacks on Linux, Windows, or any of the BSDs).

And, since you'd still need to do this multiple times in order to have multiple independent data streams, it's probably not worth the change compared to solving both problems in a new transport protocol (the way QUIC has).