A composite door can be painted, and it is harder than painting uPVC. The outer skin is glass-reinforced plastic, moulded with a woodgrain texture, and both properties work against you.
GRP is non-porous and chemically resistant, which is exactly what makes it a good door skin and a poor substrate. It needs a primer rated for GRP or fibreglass, not a general multi-surface primer, and the manufacturer's own instructions matter more than any general guide.
The woodgrain texture is the second problem. Paint pools in the grain and thins on the raised surface, so a brushed finish looks uneven in daylight even when it looked fine while wet. Spraying handles this far better than brushing, which is why most good results on composite doors are sprayed rather than rolled.
Worth remembering what the colour on a composite door actually is. It runs through the GRP rather than sitting on it, which is why the door never needed painting in the first place. You are covering a surface engineered not to be covered, and the finish you apply will need maintaining on a cycle in a way the original did not.