We close our greens each winter, located in Chicago. Time we close them is all dependent on weather; this season it was in the 3rd week of November last year it was early November. A few things are taken into account here for our decision:
1.) There are only a handful of people that would play, why should the golf course conditions be risked for a few only to hamper the experience of all the others?
2.) If greens aren't completely frozen ballmarks will occur; if greens are bentgrass they will not heal till late May leaving an open wound for Poa seed to get into and germinate during the ample Poa germination time of April through June. If you have bentgrass greens I'm sure the last thing you want is a bunch of Poa having the ability to push its way in.
3.) Risk of thaw and traffic on the greens for a few reasons; as mentioned, root sheer. Now if sheer occurs that area will not be as healthy as the rest of the green. Take the photo of foot traffic as reference, imagine half of those footprints from one foursome creating root shear; now multiply that by a number of foursomes and if root shear takes place that can set the greens in a negative direction for spring. I say "can" as maybe it won't but do you really want to find out? Second if the greens have slightly thawed and people are walking all over them footprinting may develop then those prints could freeze over night it temps drop. Imagine coming into spring with hundreds of foot prints frozen into a green that was a little too soft to be walking on. They will come out but it will take some time with rolling, topdressing etc to get greens back to that smoothness I'm sure all those that didn't play in winter desire, heck even those that did.
When dealing with Mother Nature and living things like turf there is no exact science. If there was we wouldn't continue to hear horror stories about winter kill on golf courses. This is the same thing; what is the fine line? What exact conditions will create damage and what will not; it's a gamble and the Superintendent is making the safe decision to provide the membership with the best conditions he can. Would you rather he be overly cautious during unknown circumstances or just let it go and risk your most valuable asset; the golf course, let alone the greens.
Lastly, regarding the question about foot traffic and sprayer; it's all about PSI. Now I'm not sure what type of sprayer your Superintendent has but these machines are designed to spread psi across the 4 tires very well. Manufactures know the machines need to be used in certain situations and design it to limit any damage created. Also, was there the thought that he needed to get a product out to protect the surface from upcoming weather and there was a window of time that the product needed to dry and couldn't have traffic on the turf which would risk the removal of product. I'm just throwing certain situations out there that are applicable.
Like others have said, ask your Superintendent. We are very open and willing to discuss decisions we make and help educate. Now you may not agree but that is for you to decide. Give him the benefit of the doubt; it's December and January and he is thinking ahead about how conditions will be for the membership in the heart of the golf season; May through September.