I'm no parent, but I do teach elementary school and kindergarten.
Anyway, I think it might be best if you didn't stick around the school anymore. He might be upset for a couple of days, but he needs to get used to the fact that you will not always be around. I think after a couple of days, he'll start to get more interested in whats going on in school.
Don't worry about him not wanting to "learn." He's only 6. Kids at that age learn by playing. You don't pull out a text book and say "now do these equations and then write an essay."
When I teach my kids english, most of my teaching consists of games and playing. But believe me, they have all learned so much. My kids have gone from zero english ability to being able to say "hello, my name is... nice to meet you. good bye, see you later." and can do all the colors, shapes, alphabet and a bunch of animals. All this just from playing games and having fun.
There were a couple of kids that were like your son too. They didn't like being separated from their parents and didn't want to join in on the English lessons. But now, they are fully involved and have learned so much, it just took them an extra week or so.
my 2 cents.