I've already stated that but you failed to mention the side affect of using thinner wire but more of it is reduced current capability of the secondary. Heating of the wire isn't core saturation. Heating of the wire is an I^2 xR loss. Core saturation occurs when the flux of the core cannot contribute to producing any more current.
Many negative things happen when costs are skimped for the construction/materials of a power transformer.
Keep in mind..
Transformers are also constrained in their performance by the magnetic flux
limitations of the core. Remember that ferromagnetic materials cannot support infinite magnetic flux densities, as they tend to "saturate" at a certain level (dictated by the material and core dimensions), meaning that further increases in magnetic field force do not result in proportional increases in magnetic field flux.
When a transformer's primary winding is overloaded from excessive voltage, the core flux could reach saturation levels during peak moments of the AC sinewave cycle. When this happens, the voltage induced in the secondary winding will no longer match the wave-shape as the voltage powering the primary coil. In other words, the overloaded transformer will distort the waveshape from primary to secondary windings,
causing harmonics within the secondary winding's output.
One other crucial point regarding the power transformer, internally this includes a circuit breaker and if the transformer overheats this likely will open, requiring a costly replacement.
A good stout power supply needs large capacitors and a good transformer. Skimping on either one of these components compromises the design equally.
Large capacitors help essentially the low frequency capability of the amplifier as this is when it is frequently under stress if driving full range, low impedance/sensitivity. Also the amount of regulation is important as well.
For your information in an AVR which has an SRP of $499, dealer cost of $300 the power supply components of the transformer, regulator, diodes and output capacitors will make up about 25% of the components BOM (bill of materials). I actually have the BOM for many of the popular brand AVRs and can give you specific component costs if interested, just PM me rather than taking up forum bandwidth.
I also want to reiterate once more the fact of slew rate as I feel this spec is over looked. If the slew rate is low of the amplifier section, then not even the stoutest of power supplies will rescue its performance. The signal will clip.
As I mentioned in my previous posts both slew rate & rise time are crucial specs as well the speed of the output devices. As more & more sources are capable of delivering high dynamic signals of >100dB, the weaknesses of a inexpensive, marginally designed AVRs are audibly apparent..
Just my $0.02...