This is a pretty big question so I'm going to focus on the core points.
It's not nearly so either-or, and really a range of two things: trait diversity and stability. Sheer diversity isn't sufficient to improve rate of adaptation. Stability in the sense that it's a good idea to hold on to successful traits.
In sexual reproduction, you have two sets (one set of traits from each parent) of proven survivable traits to draw from. Very roughly, it doubles the amount of positive feedback toward maximizing fitness for the environment the pair finds themselves in - of which pathogens are a major and quickly changing component.
In asexual reproduction, you have a single tested set of traits. Asexually reproducing organisms thus depend on things like horizontal gene transfer - proven traits from another organism. This has the effect of prolonging the amount of testing that takes place (one organism vs two), causing asexual organisms to be more adaptable to the circumstances they find themselves in compared to sexually reproducing organisms and vice versa.
Where I have written "proven" here, I am referring to the fact that the traits in question have survived in combination with the organism's other traits, as opposed to a totally new mutation that could have consequences before the organism has a chance to reproduce.