Would a flight consisting of solely first-class passengers be cancelled due to center-of-gravity issues? Consider any modern jetliner (narrow-body and widebody separately) in a standard 2 or 3 class configuration. If somehow only first-class or business-class passengers bought seats, would that plane be impossible to fly because all the weight is focused up front? And since they are premium-class they can't be moved aft for load-balancing.

If this is (practically) impossible, then doesn't that suggest there is a minimum number of economy passengers required as "ballast" for every premium passenger?

I suspect one answer might be that the flight wouldn't be profitable anyway with few-to-none economy seats sold, so it would be cancelled. But let's suppose in that case the aircraft needed to be flown anyway because it was required to be at the destination airport, profitably or not. Doesn't that suggest there is a maximum "premium-only" passenger count that would actually be less than the number of premium seats? I.e they would have to tell some premium passengers "sorry you're bumped, or you can go to economy" despite the plane being 80% empty?
It is much more difficult to load a plane so that the center of gravity ends up being too far forward than too far aft.
Excessive weight forward can almost always be counteracted by increased trim on the stabilizer.

When small GA planes crash due to Balance, its almost always too much rear-weight, not too much forward-weight.

Especially in the case you describe, the pax probably have luggage, and the luggage can be loaded in the rear of the plane for balance.