To compare the US and UK: In the US, the life expectancy is 76.5 years for males and 81.4 years for females. Annual healthcare spending is $14,885 per capita. In the UK, the life expectancy is 79.1 years for males and 83.0 years for females. Annual healthcare spending is $6,747 per capita. Even if the UK system is bad, it seems like the US system is much worse if they pay more than twice as much for poorer outcomes.
I think you have to account for multiple factors like Americans are generally unhealthy people, the people have better access to the top of the line medical technology, people actually spend and consume more compared to the EU. This blog post explains a lot more.
In most countries with universal healthcare, the universal healthcare provider plays a major role in preventive medicine, nudging people towards healthier lifestyles and influencing government policy on a wide variety of topics, such as food standards, consumption taxation, traffic laws, or pollution regulation. Because it's in their interests to do so, because a healthier population saves them money.
Is there the same motivation to help people live healthy lifestyles in the American healthcare system? Are there any lobby groups with the kind of influence and reach as a universal healthcare provider?
I think you inverted the causation, the provider doesn't nudge but their culture is more oriented being healthy which in turn makes providers provide the services for that. The people of the USA are generally centre right and they say that they support universal health care but the minute anyone proposes solutions to implement them, the support inverses. The same happened with ACA, it did raise costs for some people but that was enough for democrats to get brutalised for that in midterms. The solution is to implement more free Market oriented Health care like singapore etc and let people choose their options.. it would suit Americans as it isn't that burdensome compared to universal health care without private options.
The causation is correct. In countries with socialized healthcare, promoting healthy lifestyle is a cost cutting measure. So the government does it a lot, through various public campaigns and annoying laws like banning free drink refills or 300% taxes on tobacco. Usually it doesn't work, but occassionally it does, pushing the entire society toward slightly better health. Done consistently over 70 years, it naturally leads to a much more health-conscious culture, almost entirely due to government intervention alone.