What would you say the purpose or goal is of unconstructive critique?

One example would be a food critic. They give an honest description of a restaurant so readers can decide if they want to go there. The article doesn't need to be helpful for the restaurant.

But I can't think of an example where a company keeps unconstructive critics on the payroll.

Food critic is a third party. If the description is honest and contains reasoning it can be constructive to readers (potential/current customer(s) and/or the restaurant(s)). Something along the line of "sucked, bad taste" contains no substance whereas "food was burned and flavor X and Y was a bad combination" does contain substance.

Introspection.