No, that IS how it scales for range to some extent (where if you just add fuel/battery, you now need to add more power, etc), but it’s NOT how it scales for size. To first order, everything scales linearly. If you doubled the size of the aircraft, you already increased everything to compensate. This works for everything with two caveats: 1) Reynolds number means you get more efficient as you scale up in size. The effect is slight and there is nuance with respect to laminar vs turbulent flow, but it is true. 2) larger size means at some point you need truss braced wings. But this is only at pretty large sizes and only usually if you’re also shooting for high aspect ratio wings. NASA/Boeing/etc have researched truss braced wings, and it is most certainly doable.

But both of these are second order effects. And they mostly cancel out.

I am a materials scientist with a background in physics and I have also developed electric motors for electric aircraft. I have a good understanding of how the systems scale. You were mixing up the exponential range equation with simple size scaling. if it were true, then large aircraft of any type would be much less efficient than smaller aircraft of the same type. This is most certainly not true. In fact, usually the efficiency increases with scale. A 777x is much more efficient than a Gulfstream or some smaller regional jet per passenger mile.

You're describing something like the rocket equation, but that's for increasing final speed, not increasing mass.

If you double the weight of "the plane", that includes the batteries and motors.

If you start with doubling just the structure, then you might need to add 50% more batteries, not "twice as much". Then to lift those batteries it takes another 30%. Then you add bigger motors which adds even more weight, etc. etc. By the time you finish, the total weight will be about 2x your initial weight.

There is difficulty in scaling planes, but it's not because more mass requires more fuel. A fixed percentage devoted to fuel works fine, if you're assuming the same route for both planes.