Generally, a regulator must have an unregulated input to be able to have a regulated output when the load changes.
I'm sorry but that is not true. With a linear regulator you can most easily regulate to a lower voltage from any higher voltage as long as the difference between input voltage and output voltage is sufficiently high. For low drop regulators a few 100 mV often suffice, It is not the regulation or non-regulation of the input voltage, it is the input-output voltage differential that counts.
Using values too high will cause more current draw when first turned on!
A valid point. However, a few microfarads more will do no harm. There may be even more and higher capacitances to be charged from the regulated 3.3 V and 5 V lines which we do not see here because they belong to the circuit that is attached to this regulator scheme.
Those caps are for the purpose of maintaining a steady output under varying loads
Kind of. These capacitors are required to keep the regulator's feedback loop stable. A little known fact is that the output capacitor often is required to have a certain minimum ESR,
not less, for the regulator to operate stable, The theory is
explained e.g here.
and to filter out fluctuations from the ac input after rectification.
Not those capacitors shown. For smooting the rectified AC you typically use much larger capacitors. As a rule of thumb 1000 μF per 1 A of output current - although more precise calculations can be made taking into account maximum allowed ripple voltage, minimum AC input voltage etc. Again the capacitor at the input is for stability purposes, see above,