You basically just described HTTP pipelining, which has been part of HTTP for over a decade. The main problem is that somebody might have installed some broken

You basically just described HTTP pipelining, which has been part of HTTP for over a decade. The main problem is that somebody might have installed some broken proxy in the middle that doesn't understand HTTP pipelining. That is why browsers usually disable it by default. The second problem is that sometimes it is faster to open multiple parallel connections.