I have over a dozen exterior doors and they were all spray-finished before installation. Not a one suffered any damage during installation. And you will probably get a much better result overall because spraying doors in an upright position while installed is a real pain and can lead to less-than-perfect finishing. You also have to worry about masking and over-spray if they are done after installation. Follow your painter's advice...

A local automotive shop has painted our doors--some at his shop and once on-site at our home. He prefers at the automotive shop due to the dust & wind-free environment and a more even coverage. However last year we had a new prepainted door with sidelights installed. It did get marred in one spot during installation--it was heavy & cumbersome, plus it was a replacement door & did not fit the brick opening well (snug in some spots-loose in others). Then I did not like the color & had it repainted. The auto guy came and sanded it down onsite and taped off the surrounding area and it looks fine.