A private school education is not really expensive in most of the country. Think around $5000 a year. I know that's not pocket-change to many people, but you hardly need to be "upper middle class" to do it.
It'd be within the reach of more families if they could take the money that would be going towards their child in the assigned public schools with them to the school of their choice. I don't understand why that idea became a partisan one.
Most of the objections I heard were about putting that public money towards religious schooling (the interaction of government and religion being a fairly well-established as a partisan issue).
I'm not necessarily supporting the concept, but this seems a strained objection. It's equivocating the word "public." People who get social security (which comes from the government, and so is just as "public") can use it for religious causes and no one bats an eye.
And SCOTUS has approved vouchers funded via tax credits.