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The LA target should be fine for general sharpness testing, though a larger focus chart might be better in this particular case.
However, I think most people have a problem with your methodology of how you used the LA chart. If you're testing for lens sharpness, not autofocus of the camera, you need to use live view, magnified, and carefully focus (manually) on the test chart and then fire away with a remote release, on a sturdy tripod and in good lighting. Try several shots like this, refocusing between.
This eliminates AF as being the culprit, or slower shutter speeds, camera shake, etc. Essentially, done this way, the only thing you cannot control would be how the lens renders the image at 70mm and 2.8
Good luck, though looking at your 24-70 vs the 50mm prime shots, the 24-70 looks pretty good and equivalent to the prime.