After some trying I found out how to fix the ‘fingerprint’ pattern. It looks like changing the ‘distance’ in the shader editor. A distance value of 1 gives strange patterns. Lower values don’t have this; see the images below.
So, now it remains a mystery why the images are still so ‘pixel’ looking. Does anyone know why this is? After looking a lot on the internet I found that it might have to do something with the viewport resolution. So I tried to paint on the UV map instead; unfortunately that gave the same results.

