Congrats to Jay for his first invite. How exciting for him.
I throw large parties all the time but they're almost always family gatherings where everyone pitches in. I honeslty can't imagine throwing a birthday party with that many kids!
The rule at the girls school is birthday invitations have to be mailed not handed out at school no matter who you're going to invite. This works well for us. My kids generally get along with everyone but they're not necessarily close friends with all their classmates and may not want everyone at their party. Plus they have friends from activities outside of school so if we invited all their classmates and all their other friends we'd be looking at 4 parties a year with at least 50 kids each! No way! We just can't afford that.
We let our girls invite up to 10 of their closest friends (it usually ends up being less than that) and we try to keep things simple. They're spoiled enough as it is by all of our extended family.
In your situation though, if you're up for a big party and if he wants all his classmates there then yes, absolutely order pizza, get a sheet cake and have it at a park! It sounds like it would be fun.