I think valuing the hospital comes the novelty maybe?

I’m no expert on this topic, but I’ve previously read that when a thing is made free people stop valuing it.

I read that as well and had trouble finding it but this sounds about right:

Why not make the fares free in Queensland? One reason can be found in the experience of the Miami Beach Transportation Association in the United States. The Association launched free shuttle buses along the coastline. However, the lack of fares led to a diminished sense of responsibility for the upkeep and care of the transit system, ultimately negatively affecting both driver satisfaction and passenger experience. Whilst passenger numbers initially surged, studies show problem riders resulted in raised personal security concerns as transit crime increased. Examples include increased assault, damage, and theft for users, becoming a deterrent for both new and existing riders. An attempt to resolve these issues was introducing a $0.25 flat fare, leading problem riders to avoid the service. Consequently, these negative factors began to rapidly decline, such as vandalism decreasing by 90% whilst passenger numbers remained steady.

https://ninesquared.com.au/insights/nearly-free-fares/

I'm no expert on this topic, but I've previously read that when a thing is made free people stop valuing it.

This sounds like a misunderstanding of economics. If someone gives away something for free, they're only saying the thing has no value to them, etc, or — in the context of gov services — the act of giving it away has more value. It's never that the thing itself has no intrinsic value, "period".

You could definitely argue that the 50c fee prevents homeless people from squatting, or completely unnecessary travel, but I would argue that everyone should be provided with shelter, and 50c probably isn't going to prevent unnecessary travel anyway.