It can in fact become a worse lens on crop! The smaller sensor has more lines pairs per mm and so you have to look at the higher level MTF values of the lens with the crop to get the equivalent for the FF. For example, the 30 lp/mm MTF for the crop is equivalent to the 20 lp/mm of the FF, and the 48 lp/mm MTF for the crop is the equivalent of the 30 lp/mm of the FF for the overall resolution of the sensor. So, a lens that has a very good value of the 10 lp/mm and OK 30 lp/mm but deteriorates rapidly with increasing lp/mm will be OK for FF but weak on crop. The high quality lenses like the 300mm f/2.8, 100-400mm II perform well on crop. My old 100-400mm Mk 1, a poor copy, was quite good on FF but rubbish on crop.
No, the quality of the lens does not change. The optics are the same.... in fact, on a crop body you are only using the sweet spot in the middle of the lens so the overall performance should be better.
What does change is that crop sensors are (usually) higher pixel density, and as such need a higher resolving lens to achieve the same apparent sharpness as a FF body, so yes, the image will appear fuzzier than a similar image on a FF camera taken with a lens of similar sharpness..
That said, DXO does not rate the lens in line-pairs, they rate it in megapixels of sharpness....
And here is where it gets real silly fast. Take the 7D2 and the 5DS R. Same lens on both cameras. The pixels on the camera are the same size, yet the poorly named "sharpness" metric plummets. In reality, this so called metric is not as much about lens performance as it is about the number of pixels on the sensor.
We end up with a metric that can not even be used to properly compare the same lens across two cameras in the same brand! How could it possibly be used to compare between two different brands? Or even worse, two different lenses on two different brands!