I can't speak for Denmark, but I can speak for New Zealand, which has similar spend, and similar results (compared to the US). I wrote a thesis about healthcare software in uni, but as part of that I interviewed a bunch of people across the NZ healthcare industry, and learnt a lot about it.
A big efficency for NZ is 'pharmac', which is essentially a country-wide medicine subsidizer, and supplier. It's a government entity which figures out how much of a certain drug the entire country will need for say a month, then goes and buys it, in bulk on the international market. Making such a large purchase gives them really good bargining power, and they'll buy generic versions. Not all drugs are pharmac covered, but most are. It means there's not much profit margin for drug supply in the system.
The system is setup into 'district health boards'. Their job is to manage the health of everyone in a geographic area, and this means both in hospitals, and in primary care (your local doctors office). Because of this, they see the cost of their entire system, and are incentivised to optimise the care overall, as opposed to say a private hospital, and a private doctors office optimising their individual businesses.
One way this manifests, is putting a big focus on getting people to go to primary care, instead of hospitals. Hospitals are the most expensive thing in the system turns out, so they'd much perfer people get to a primary care doctor sooner, before something becomes an issue that requires hospitalization. So they're incentivised to keep doctors visits cheap, and accessible. This comes into play quite a lot with elderly care, they make up a huge amount of primary care visits - but if they don't become hospital visits, that's great.
You mentioned drug use. That's an interesting one as that's a government-wide issue, not just a healthcare issue. It affects poverty, education, emergency housing, the economy as a whole. There's been a big focus on public education, and actually ad campaigns about tabacco and alcahol, trying to make them seem really un-cool. And it's sort of been working, smoking rates are down, I'm not sure about vaping rates though. Meth is a big issue in NZ, but opiate/painkiller abuse is uncommon (compared to the US).
One general point though, is there's not much profit oppertunity for companies in the system. As the majority of money comes from the government, the funding is very standardized. Running a doctors clinic or say a pharmacy is just like running a small business, with the same oppertunities. There just isn't huge 'startup' oppertunities to find a killer product and turn a huge profit. Profit isn't taken at every point of the supply chain in care.
NZ's system has flaws, but overall I'm really pleased with it, and glad that we have it.