There is never a reason to leave the holes open in the greens after aeration. There is three main reason to aerate and all of them require the holes to be completely filled with sand. Do a google search and there are thousands of articles on the reasons and methods of doing a successful aeration. The USGA website has a number of articles pertaining to this subject and the importance of doing a good job of it to heal the greens as quickly as possible. If the soil temp is warm enough the greens should heal in no more than two weeks and if the soil temp is not warm enough the aerator should stay in the shop. If some guy is telling you that it is good for the greens to leave the holes open he doesn't have a clue and should be fired before he does any more damage to your course. I don't know where these ideas come from because they don't come from any science or research that has been done by any reputable organization. Greens have been aerified for many, many years and there really isn't anything that is done that keeps the greens unplayable for as long as possible. I would not be happy to belong to a golf course that does things that make the course crappy when there are techniques to use that put things back in shape very quickly. Good luck, you are going to need it.