IIRC there are still post-operative cognitive problems in major surgeries done without general anesthesia, suggesting that there's something else going on too. (I don't doubt that anesthesia plays some role, though.)

edit: Found the reference: https://www.frontiersin.org/articles/10.3389/fpsyt.2018.0075....

"More convincingly, a recent study demonstrated that open abdominal surgery under local anesthesia caused increases in hippocampal IL-6, TNFα, and memory impairments (71), suggesting that anesthesia per se is not necessary for the production of neuroinflammation and subsequent development of POCD."

"As seen in observational studies, a prospective randomized clinical trial comparing the use of general vs. spinal anesthesia in extracorporeal shock wave lithotripsy showed no significant difference in the incidence of POCD defined by a neurocognitive battery (142), suggesting that surgery and not anesthesia causes POCD."