That taxpayers pay for government programs isn't the big revelation you seem to believe it is.

Yes, it actually is, to many people.

Poll a random subset of people with the question "are you in favor of free childcare?". X% will say yes.

Poll another set with the question "are you in favor of taxpayer funded childcare?". Y% will say yes.

I would bet any amount of money that X>Y, and (X-Y)% of people did not think about the fact that a free government service is not actually free.

Exactly how big X and Y are, I couldn't say. But identifying propaganda and deceptive language is never something that should be discouraged, even when it's advocating for a cause you agree with.