For example, countries with socialized healthcare have significantly more walkable and bikeable cities. That’s not a coincidence.
The most plausible link would be if people couldn't afford cars under a socialised healthcare system; but I doubt anyone is going to try and argue that seriously.
No. It does not. Not "suddenly" or "magically".
What does happen is that social health concerns and advice feed back into other public policy making decisions.
Advertising agencies get contracts for campaigns to improve health awareness, walking and biking paths become routine considerations in city planning, etc.
This takes decades to iterate through from non existent to commonplace.
You might like to check whether the layout of the citys involved were decided on before or after the invention of the car - and the introduction of universal health care - before you post anything. That'll probably come up.