We fully support increasing access through the expansion of licensed child care. However, I would disagree with your premise that it's not possible to make licensed child care accessible to all without resorting to funding a for-profit sector.
We know from other countries that, in fact, it is very possible to achieve a universal child care system of a very high quality exclusively through expanding, in a very deliberate and conscious way, the provision of not-for-profit and public child care.