If I'm understanding the "inflammatory milieu generated by surgery" part correctly, does this imply that the cognitive effects would be equally likely if surgery were performed without any kind of anesthesia? (Or to put it another way: the anesthesia isn't directly implicated, it's just that anesthesia and surgery tend to go hand in hand...?)

Right, there are operations that are sometimes performed with minimal (or sometimes even 0 sedation) with just a nerve block (regional anesthesia) which impairs sensation locally (like you get when a dentist numbs up your mouth, but you’re still awake). They have the same rates of cognitive dysfunction afterward.