There is a flag to install a snap in classic mode, which I think would allow it more access, but I've not seen more fine-grained control than that. Perhaps it exists, I have not had a reason to look.
As for X11, I'm on Wayland and wouldn't expect xeyes to work. What X11 programs still is used seems to work fine in Xwayland, but I don't use VNC so I don't know how they interact.
So I don't know if these are solved, sorry. I suppose I have simpler needs.
Part of that is much of the point behind snaps (and just as true of flatpack for that matter) - once applications are put into sandboxes many things break. And while it is often easy to erect walls against unwanted use it takes a lot of thought building abstractions that let people do what they need in a controlled way that don't leave the system as open when compromised.
But yes I agree, it is definitely a big change compared to how things traditionally have worked. The question is if it is a necessary direction.
Well, if access to my filesystem stops working as expected, then I'm sure many people would agree that is not a desired direction ...
Linux has traditionally used the model of: users are suspect, applications are trusted; of course you can ask questions about that, but don't flip it upside down while breaking stuff. For example, I would probably be OK with it if they added a popup asking me to set permissions. But instead, it just breaks.
As for X11, I'm on Wayland and wouldn't expect xeyes to work. What X11 programs still is used seems to work fine in Xwayland, but I don't use VNC so I don't know how they interact.
So I don't know if these are solved, sorry. I suppose I have simpler needs.