It isn’t clear to me from the description how this snap specifically works and how it differs from logmein-host. The snap for logmein-host does not require classic access and, looking at the snap.yaml for it, it seems to run daemons (which all start as root) and presumably a client can connect to it and have ‘root’ access to the filesystem areas allowed to the snap (ie, the snap-specific directories).
For x11vnc integration, it is unclear what you are trying to do in terms of implementation. Are you assuming there is an X server running? If so, you are correct that you cannot use su to run x11vnc under that user. Better would be for you to use a user service within the user’s session that is launched automatically upon login, and then connect to that. Today, user services are still a work in progress, but you can use The snap format in the meantime.