For traditional laminated EI mains frequency power transformer cores, the core dimensions and core material determine, for a given primary voltage and frequency, a minimum number of primary turns required to prevent core saturation. There is of course some wiggle room in deciding what level of flux defines "saturation", but as a first approximation, the original claim is true.
Once the number of primary turns is determined, the VA capability of the transformer is limited by (among other things) the winding ampacity (current carrying capability). Therefore, classically, transformer designers would use as much copper as would fit through the core's window to minimize copper losses, and heating, and consequently maximize VA rating. Typically, half the window space would be filled with copper for the primary winding, and the other half would be for the secondary winding(s).
However, I have noticed that this design principal does not seem to be applied to toroidal mains frequency power transformers. That is, the core window tends to contain significant area that is not filled with copper. For example: taken from here.
Obviously, there are physical interference problems with adding more copper. There is a given wire thickness such that number of primary turns will completely cover the surface of the interior hole of the toroid. But there should be another (larger) thickness such that two layers, (each with a different number of turns) would have, in total, the required number of primary turns. Or alternatively, there could be two layers with different diameter wires, each with the required number of primary turns, which are wired in parallel. Yet a third possibility, is to have one layer with the required number of primary turns, a second layer with almost that number with the remainder forming a sparse third layer.
It seems to me that the mechanical problems are increasing the copper through the window of a toroidal transformer are not insurmountable.
My question is, given that a higher price may be charged for a transformer with a higher VA rating, other things being equal, why do manufacturers of mains frequency toroidal power transformers leave such a large area in the window of the core empty of copper? Is there some other limitation on the VA rating of a toroidal core transformer such that adding more copper would not increase the VA rating? Is there some other factor, such as flux leakage, that would sufficiently offset economic advantage of higher VA rating? Or is it simply that the technical challenge of adding more copper would offset the price advantage?