But why would a strong, inheritable trait that cuts fitness by half not be selected against?
Well, I suppose if you break it down, the question is regarding:
A trait that is possible to be inherited, is even favorable against other traits, but not a strictly dominant trait (otherwise “dominant trait” would be used instead of “strong, inheritable trait”).
This trait reduces overall fitness by one half (the trait is maladaptive to the respective environment).
Natural selection has not driven it out of the gene pool (when, ostensibly, such a maladaptive trait should have been driven out or at least not be potentially more and more pervasive over time; even if that “strong” means the trait is favorable slightly over stagnation).
So why the discord?
Hopefully that helps; in terms of the biological answer, I think more context is needed – but I can think of examples where this happens…