And then the window manager spawns whatever programs the user wants, and they end up sharing that $DISPLAY.
1. There is nothing fundamentally insecure about the design of X that couldn't be fixed with a bit of effort.
2. The effort required would be significantly less than what has gone into Wayland over the last 16 years.
Fixing the security of the system certainly would involve changes to more than just xorg. Anywhere there is a security boundary you'd need to make modifications - but that's OK, most of these security boundaries are younger than wayland anyway.
This discussion of $DISPLAY seems like distracting minutiae. Sure it's a medium size problem that would need solving, but that's all it is.
To address your specific point: it could be the responsibility of the window manager to set $DISPLAY to a less powerful socket when starting processes. Ultimately it doesn't matter though, because if the display server isn't doing some sort of sandboxing then the spawned process can just ptrace xorg and do whatever it wants. X being secure only matters in the presence of a security boundary and in that case it would be the responsibility of whatever is setting up that boundary to create a less privileged socket. Whether that be ssh or flatpak or whatever.
Once again, for emphasis: nobody has stepped up to do that, in the history of X (outside of proprietary products, of unknown quality).
Wayland is not some outsider project competing against X11. It's the people who were developing Xorg, saying they're fed up with X11 and need a clean slate to fix its problems.
Here's my unpriviledged user trying to ptrace my Wayland compositor running as me:
strace --syscall-limit=1 -p 697351
strace: attach: ptrace(PTRACE_SEIZE, 697351): Operation not permitted
In a world where the people doing the work switched to developing the Wayland ecosystem, arguing about a "could" that requires changing every window manager seems like a guarantee X11 will not get improved. Feel free to put effort into it...