First question: can soap ingredients have any effects at all due to soap being a wash off product?
Based on many reports of people actually trying a soap and finding that it is good for the skin or instead irritating, based on the fact that there are medical soaps out there, I personally believe that soap ingredients have
a chance of doing something
On the downside, it seems that only a small number of substances are able to penetrate the uppermost skin layer.
For flavonoids, I go by colour: if the colour is stable, likely the chemical structure remains the same.
Carotenoids, I believe they are stable - I vaguely remember reading somewhere that sodium hydroxide is used for their extraction.
Anthocyanins do change colour, but unless I am wrong, the colour reverts when pH is lowered. Which means that they are not damaged, but the question remains wether they are active in their icky brownish state.
There was a site out there talking about which EOs are lye reactive, and I believe that tea tree is not. Again, the Duck law: if it smells like eucalyptus, it probably is eucalyptus.
Proteins, I just read somewhere that they are split to amino-acids, which isn't necessarily a bad thing.
The activated charcoal still has a large surface* area inside the lather. But I am not convinced that it can extract toxins from inside the skin. I would guess that it physically neutralizes the chlorine in the water - can't say if to a relevant degree. It is also mildly scrubby...
*Not sure if that applies to your monkey thingys, since from the picture, the coal seems to form pin-sized little clumps. But maybe it's more scrubby that way