HTTP isn't only about NAT. Lots of protocols have a clear notion of client and server, and don't require dialback.
HTTP is successful partly because it's simple and "stateless", in the sense that all protocol state is held by the client in cookies. Keeping all state in cookies+database provides some useful properties, in particular around scaling, being unfazed by IP address changes (e.g. wifi to wired), reboots of both clients and servers, version changes, hibernation/wake-from-sleep, and a variety of other things that otherwise break connection oriented protocols.
Of course sometimes people screw that up in other ways, hence the popularity of "clear your cookies" as a troubleshooting tip. But overall it works pretty well. Look at the prevalence of hacks on top of SSH to create persistent sessions, to see what has to be done when you don't have that model. SSH only survives as a protocol because remote administration doesn't need to be a particularly great user experience and it doesn't need to scale, so legacy and inertia wins out. Otherwise we'd probably see a request/response protocol take over there too.
X11 was always a PITA not only due to NAT but also due to institutional firewalls, e.g. universities often have enough IPs to give everyone a public IP but firewall traffic anyway to stop exploitation of vulnerable devices. So the whole "remote client connects to local server" model was not only confusing but also a poor fit for the real world internet in which centralised, professionally run servers are invariably more secure than scattered non-professionally run edge devices.
HTTP is successful partly because it's simple and "stateless", in the sense that all protocol state is held by the client in cookies. Keeping all state in cookies+database provides some useful properties, in particular around scaling, being unfazed by IP address changes (e.g. wifi to wired), reboots of both clients and servers, version changes, hibernation/wake-from-sleep, and a variety of other things that otherwise break connection oriented protocols.
Of course sometimes people screw that up in other ways, hence the popularity of "clear your cookies" as a troubleshooting tip. But overall it works pretty well. Look at the prevalence of hacks on top of SSH to create persistent sessions, to see what has to be done when you don't have that model. SSH only survives as a protocol because remote administration doesn't need to be a particularly great user experience and it doesn't need to scale, so legacy and inertia wins out. Otherwise we'd probably see a request/response protocol take over there too.
X11 was always a PITA not only due to NAT but also due to institutional firewalls, e.g. universities often have enough IPs to give everyone a public IP but firewall traffic anyway to stop exploitation of vulnerable devices. So the whole "remote client connects to local server" model was not only confusing but also a poor fit for the real world internet in which centralised, professionally run servers are invariably more secure than scattered non-professionally run edge devices.