The important current to consider in your transformer is the average current.

Most certainly NO. Transformers are not wound for average, but for PEAK current. You are aware that transformers for resistive loads are wound for a higher Bmax than ones used with rectifiers, and in fact the typical value for toroidal transformers for resistive loads like halogen lamps is around 2T (and note, the lamp has a VERY low resistance at startup so i am not talking abut the few startup cycles!), while a typical transformer wound for a rectifier application is would for about 1.7T. Where do you think this value came from? A rectifier with a capacitor filter is NOT a resistive load, so neither is peak current directly related with the load, it does not even occur in phase or indeed for the whole duty cycle (hence has to charge the capacitor enough to cover the whole duty cycle). It might surprise you it comes from a TABLE pre-calculated for a certain filter capacitance per ampere of load current - calculated from the maximum PEAK current expected from that setup. And it's obviously not SQRT(2)*RMS, but rather higher.

Homework: put a small winding on a loaded transformer driving a load through a rectifier and filter - then look at the secondary voltage with respect to load current and filter capacitance. At some point the tops will become flat. What do you think happened there? True, as long as it does not happen, you are mostly fine.

Why mostly? Because the shape of the peak current does have some added effect due to iron losses. A shape with less high harmonics will generate lower losses (And keep in mind they go up with harmonic frequency squared).

Ever designed a switching power supply?

Have a look at what happens due to reverse recovery and how it relates to PEAK current. Yes, the average current is to the first approximation indeed the same, but that's not what causes these problems. Note: absolute maximum current for the peak is dependent on reverse recovery, because during reverse recovery the diodes are sucking out the charge out of the capacitor, the longer the higher the peak was during the proper direction. So, more peak current, less remaining voltage. Of course, it comes down to properly choosing diodes, and the question, are they indeed?

For the capacitors, they need to handle the ripple current required for the amplifier's current needs. Average current will not be affected by different capacitance.

Granted. But what current heats up the capacitor's internals? Average current of the power supply?

What calculation do you use to select your power supply capacitors?

Simulate it. Starting form a standard transformer calculation for some 1.7T or, depending on experience, knowing in advance some of the parameters (such as very large filter capacitors) lower, then depending on peak vs average current you go back and adjust the transformer specs (in case you are doing a custom transformer), or adjust the filters if you have an off-the-shelf transformer. Warning: most diode models do not simulate reverse recovery. So, some use of the graphs in the diode datasheet is to be expected.