I think that analogies are helpful for elucidating the point but in terms of concrete evidence, there are two gold standard studies that really reveal this issue. These studies are very hard to come by because it is typically difficult (for good reason) politically to experiment with people's healthcare, but we are lucky to have two: the RAND healthcare study and the Oregon medicaid lottery.
My understanding of both of those studies is that (particularly for pre-registered analyses), we saw that adding some sort of cost-sharing substantially reduced utilization of healthcare services (~30%) without any impact on health indicators even multiple decades down the line, with the possible exception of mental health indicators. Nowadays people try to p-hack their way out of these conclusions, but it is pretty strong high-N experimental evidence.
My understanding of both of those studies is that (particularly for pre-registered analyses), we saw that adding some sort of cost-sharing substantially reduced utilization of healthcare services (~30%) without any impact on health indicators even multiple decades down the line, with the possible exception of mental health indicators. Nowadays people try to p-hack their way out of these conclusions, but it is pretty strong high-N experimental evidence.