There's a thought provoking thread about the sharpest Nikon lens at f5.6. There's no qualification about across the frame sharpness being required or not or of edge or corner performance but the op does go on to say they don't care about bokeh, vignetting, distortion, field curvature and ca and anything else aren't mentioned.
If looking at variable aperture zooms sharpness I suppose sharpness at f5.6 (wide open at the long end) could be an issue but for primes and f2.8 zooms I'd expect them to be well into their stride at f5.6 and perhaps sharpness wont be an issue for me at least in the central area but performance towards the edges and into the corners could be under question as could all the other things such as ca, distortion, field curvature etc. and bokeh. And of course if sharpness is the only criteria some lenses are at their peak before f5.6 but of course using them at wider apertures where they may be sharper could have implications for DoF or even across the frame performance.
I can't think of a lens I've had that caused me to worry about sharpness at f5.6. Maybe the Sigma 28-300mm I had but the main reason for owning that lens was the do it all zoom range so I never expected sharpness to be all that good and acceptable for whole image viewing would be fine

These days I just can't think of a lens I'd criticise at f5.6 and even old film era primes such as that Nippon Kogaku 50mm f1.4 are IMO plenty sharp at f5.6.
Is sharpness the overriding factor for you at mid apertures or would you balance that against other qualities and issues?