I actually had to use DeX in an "emergency" and I was impressed how well it worked. Connected to the work VPN, connected to my workstation via VNC and got some work done. Everything "just worked", I was shocked at how smooth the entire experience was.

It's still there. It's not past tense.

My phone only supports outputting to one external screen (unless you count mirroring or the phone's screen)

The most obvious limitation to me is none of the browsers really adapt to being on a desktop (middle click open new tab, multiple windows, dev tools).

Is it "Desktop mode" as in Android on the desktop - or is it Desktop Linux (like KDE or Gnome) backed by some distro flavour?

Android on the desktop.

The only desktop exclusive app is the shell itself. Otherwise it's all the same apps as you'd be using in phone mode.

The shell is Samsung's own thing. But it works well enough- it has a start menu like launcher. It has floating windows (like every mainstream desktop os), but if you launch too many older apps disappear. You can switch windows with alt tab, using the window list at the bottom of the screen or clicking behind your current window.

The only other party that's capable of pushing this kind of functionality onto the industry is Google, and Google does not have HDMI out on any of their current Pixels. Perhaps it's intentional to prevent Android from competing in the same market as Chromebook.

I suspect this is more a software ecosystem problem.