Living near Boston and being a Celtics fan too (they're my number 2 team), I have thought of this exact trade a number of times. Someone even called into a sports talk radio show out here and proposed it. And for all of the same reasons (from the Cs perspective) as the OP.
I like the trade for both teams, but I would also be fine with keeping Hayward in Utah. Assuming he does bolt in FA, I do believe he will go to Boston, especially given Danny Ainge's prior interest in him and the fact that both Ainge and Stevens just re-upped their deals with the organization . Given that, Hayward for #3 would be a good get for Utah.
Don't know that maxing Barnes is the way to go, but we do certainly have to account for Burks being so fragile. But yeah, Hayward for #3 might be the way to go.