We have three government levels, State (the whole country), counties, and municipalities. Healthcare is mostly handled on county level. It works well enough I think, but there are probably better ways to do it. I don't think most people know enough about it to say whether it works good or not.
There is nothing terribly wrong with the system itself, but there are a lot of small things that could be changed and improved. Most notably I believe dental care should fall under regular healthcare instead of being it's own thing. The fact that we have a cap to how much one person has to pay every month is very important.
There is a difference between elderly care and healthcare. They are two completely different organisations. General elderly care is done on municipal level in either elderly homes or in their private homes. But hospital healthcare is handled the same way for everyone. I think there are lots of cooperation between the two when it comes to the elderly, but I don't know enough about it to say anything specific, even though I've worked in an old peoples home. It has nothing to do with age btw, just extra services for those who need it, like help with cleaning, food, transportation and so on.
The main concern people have are over how available health care is, queues and so on (shortage of healthcare workers is a big issue). Then there is obviously a big political discussion over how financially inefficient it is run and how much the government should actually spend on it, and ways to make it more efficient and so on (privatisation mostly). The general system isn't discussed much from what I can tell, so I think people are fairly satisfied with that.
I personally think mental health care could use some changes, from my own experience. But that's pretty specific and not something most people think about.