Glen Cove is about 15-20 minutes from the academy. You need a rental car if you are staying there, but parking is free. The grounds are beautiful, the service is good, the food in the upstairs pub is excellent despite a limited menu, the rooms are okay but not great. We have stayed there several times and have enjoyed being more out in the country rather than crammed into the city. Our DS also enjoyed getting away from the school area on Parents'Weekend. If you make reservations early and negotiate, you can get rooms for around $100.
I presume the Glen Cove hotel you are referring to is The Mansion
http://www.glencovemansion.com/
... I believe we were eating breakfast with ladies who are employed in one of the worlds oldest profession. Pay the piper and stay in Great Neck.
You probably were indeed dining in the company of "professionals" and I agree it's best if you let your Mids save that experience for their sea year...
There are options for your DS/DD going solo to Indoc. Often, PC's can meet up and share a room. Sometimes a new family will add your DS/DD to their group. Our DS was 17 for Indoc and could not rent his own hotel room. We contacted Admissions and they matched him up with a local family that picked him up at the airport, took him home, fed him dinner and breakfast, and took him to the academy to report the next morning. We were pretty overwhelmed by their generosity. We have since come to know that things like this are not unusual and it's all part of being in the King's Point family. You just have to ask and help will magically appear!