Not sure I agree with this, or we're talking about different things. Why would a DX lens only be using the sweet spot, it uses the full lens (near enough) just like FX on a FF body.
Simply this: yes, lenses are always less sharp towards the edges of the field, and the larger the format, the greater the fall-off (very generally). With DX lenses, the designer doesn't have to worry about the extra coverage outside the APS-C image area, but the higher sharpness closer to the centre is still there. The best DX lenses are very sharp, with less fall-off across the frame, even more so with some M4/3 lenses.
now this is something that I'm struggling to get my head around. I understand that if you take an image and then crop it you're losing resolution and therefore sharpness, but what I don't understand is why crop sensors put more demands on a lens? I get that it's more 'zoomed' in but don't understand why this would affect resolution and sharpness?
(Not arguing here by the way, genuinely interested)
Contrary to popular belief, pixels are not a major factor here (within reason). 'Sharpness' is about the relationship between resolution (the fineness of detail) and contrast (how clearly those details are rendered). Of the two, it is contrast that plays the bigger part in our visual perception of sharpness. Resolution and contrast are the two axes of an MTF graph (Modulation Transfer Function - your link) and there's one fundamental fact of physics at play: as resolution goes up, so contrast goes down. See any MTF graph where performance at different resolutions is shown.
It's like a car that accelerates from 0-60 in six seconds, but takes a heck of a lot longer to get from 60-120. The more you want, the harder it gets.
With full-frame vs APS-C (FX vs DX), FF has to be enlarged less for a given size output/print. The difference is the crop factor, and in resolution terms, it means that to achieve the same standard of sharpness the APS-C lens has to deliver the same level of contrast at, say, 36-lines-per-mm as the FF lens does at 24-lpmm (24x1.5=36). That doesn't happen, and that's why FF is sharper than APS-C, which is sharper than M4/3, which is sharper than smaller formats etc etc.