People keep getting stuck on this #1 center thing. What the Leafs needed more was a defensively responsible center... You know to help stop a team from mounting a comeback in a game where the team scored more than enough goals to win. In fact the Leafs were one of the highest scoring teams in the NHL with Bozak as a number 1 center.
It's just that he doesnt have the tools to be a great #1 C or even a #1b kind of guy.
he's an offensive black hole, someone in another thread figured out he had 8, yes 8 primary assists this year, that's BRUTAL when you're playing with linemates that scored 38 goals in total(add a few in from lupul too).
EDIT: Meanwhile a guy like Kadri had 21 primary assists this year.
Bozak rides the coattails of his linemates, he's a decent player, but not someone who makes the 1st line better, he just coasts on it.
his defense is average, maybe above average.
Seriously though, I've been watching him for a while, he doesnt do anything special on that line outside of faceoffs and the odd pot in goal or decent play.
get in a guy like Stastny(who does everything you listed in a better way, or at least he used to) and you'll see a massive overall improvement to that line.