I cannot understand that such a vital piece of info is not foremost in training...
Not that simple. Overwhelming majority of go-arounds are performed or at least initiated before tochdown, with the pilots realizing the need to abandon the approach. Pressing TOGA is trained to become the instinctive reaction. We know of two fatal accidents (S7/TAM) where inadvertent selection of above idle thrust on ground caused an overrun, so the logic inhibiting the TOGA with WOW is entirely correct.
The fundamental issue to address here is that there was no need for the go-around in the first place, so the crew was not mentally prepared for executing one following the touch-down. A key contributing factor was the adherence to SOP after the confusing and in the circumstances totally unnecessary system warning.
I would be much more comfortable with a warning related to RWY length remaining, with a predictive logic so the warning could be made while still in the air if the aircrafts drifts too far past the threshold. Another change in logic could be that such a warning which by SOP mandates a go-around should also re-enable TOGA even if on ground if T/R not yet deployed, enabling the much faster instinctive action to be performed rather than relying on the slower cognitive process