My hill climb PR's, and by that I mean long, steep gradient climbs, are all from riding my 5.8kg Wilier, a bike with aero wheels and forks but otherwise optimised only for weight reduction. It accelerates very quickly albeit not a great bike for powerful sprints but great for climbing; super stiff, ultra responsive.
It is a very different ride with a night and day feel compared to my heavier aero bike with which I definitely sprint and TT better on; Strava times prove this. While obviously the engine is the single most important aspect of cycling, it is false to claim that the bike itself makes no difference. Like most, I am demonstrably faster on a lighter bike than a heavier, aero bike on steep, long climbs local to me - circa 6 - 20+%, 8 - 24km. Same engine, different bikes, each optimised for different goals. The time differences aren't big, they are small, sometimes down to a second or two, sometimes nearly a minute. Enough to make a difference if you chase such fine margins.
I have a compact 50/34 on both, as is my preference nowadays.