"The Change for Children would be, at best, a component of the BOE, which itself is embedded in a much bigger system."

They already have been. And one of them voted to raise the taxes, and is now running on the opposing slate, while another remains on the Change for Children slate.

"But the opposing slate has said, essentially, that they don't really think the budget should be a major consideration, that they have no problem with raising taxes, and that they don't really believe in any measure of accountability."

This is simply not true. I spent the past several days reading interviews with the candidates both together and individually, as well as reading coverage on their debate, and they have said the exact opposite of what you are claiming here they had.

"So I would prefer a BOE that was not entirely union,"

I agree. And it's not. Also, important to note that being Union and being backed by the Union are 2 different things.