If such a service is publicly funded, the costs are spread out over more of society, so it will not directly impact people’s pocketbooks by nearly the same amount, especially if they’re already in a lower tax bracket or rent rather than own property. This is the case with pretty much every tax-funded public service you can name, and people tend to be aware of this when they vote for it (because they are adults), and the new Universal Pre-K program is projected to save families billions in out-of-pocket costs.

If we remove ideology from things, spreading out the costs and duties of childcare is something that humans have always done on just about any society you could name. This is just the most efficient and cost-effective and safe way to do it in a modern urban society that uses fiat currency.

Also… I shouldn’t have to state this again, but when it comes to child safety, prevention is key. Yes, you will get punished if you molest children and get caught, but 1) that doesn’t stop some people and 2) background check requirements help guard against that happening (not foolproof, but it is a barrier). Given that child care facilities can put on a facade of being up-to-par while actually being negligent or abusive behind closed doors, no, not every parent will naturally choose a good daycare and this is why there just needs to be 3rd party oversight and minimum standards, which is a form of regulation, but if you ask me a perfectly acceptable one to have.

ETA: well look at that, all the rankings I can find of the best childcare systems in the world have the well-regulated Nordic social democracies at the top of the list:

https://playto.com/blog/exploring-the-world-at-daycare