tightvncserver starts up just fine using :1, :2, etc, None of the control the default X Server on the monitor itself.
If I change it to :0 I get this error A VNC server is already running as :0
If I change it back everything works just fine.
Does tightvncserver not run the native display?
No, it is not possible with tightvnserver. Most vncservers will try to build a whole WM (=window manager), while your :0 display has, apparently, all of this.
You can check this by not starting your X server at boot, and before issuing the usual startx command, to start tightvncserver on :0. It will work then.
On the other hand, if you are keen on using display :0, you can try x11vnc, which is what I do.
Try x0vncserver. i.e. On arch:
x0vncserver -display :0 -passwordfile ~/.vnc/passwd
Install
$ sudo apt install tigervnc-scraping-server
Setup password
$ mkdir -p ~/.vnc
$ vncpasswd
Password:
Verify:
Would you like to enter a view-only password (y/n)? n
Starting the VNC server
$ x0vncserver -passwordfile ~/.vnc/passwd -display :0
or
$ x0vncserver -localhost no -passwordfile ~/.vnc/passwd -display :0
