Controlling for interaction effects
I leave for another thread the issue "What is the justification of using the arithmetic operation of multiplication and so the product of two variables in order to model the concept of interaction?"

Practically, product-terms come from the tradition of the Translog regression specification, which is a second-order approximation to any actual specification, and produces such products.
They are therefore considered "second-order" effects, meaning that they are included in a specification alongside the stand-alone variables. This means that their importance is judged after the effect of the stand-alone variables has been taken into account.

I have not seen in the literature a situation where "interaction terms" are included alone, without the stand-alone presence of the two variables forming the product.

So for the specific study mentioned by the OP, conventional practice would ask "why not include also interaction terms?"

Nevertheless, it appears intriguing to argue convincingly that in a certain case, we should only include the "interaction term" of two variables.