I agree entirely on the "power loss" thing being a load of BS, especially within the range of flex we are typically talking about in practice.
But that doesn't mean stiffness is not important in several other ways i.e. handling and comfort. A frame with poor torsional stiffness will handle like a soggy couch. Bars that are noticeably flexible will not provide solid support when sprinting hard in the drops or climbing out of the saddle. Too much fore-aft flex at the headset and fork can make braking very unstable (try riding with a slightly loose headset and see how that feels when braking hard on a steep downhill). The best frame designs optimise stiffness in specific load directions i.e. combining high torsional stiffness with relatively low vertical stiffness (e.g. D-shaped seatposts, super thin seat stays) to give both precise handling and a reasonably compliant ride. Wheel stiffness, both lateral and vertical, is also important for handling and comfort.
Stiffness certainly matters a lot, just not in terms of power delivery.