The "the US is too big" argument gets dragged out for all kinds of things, and it is total nonsense. You have more people to pay, but you have equivalently more jobs. And conveniently, you have administrative subdivisions: States. And not that many of them have more people than the Scandinavian countries.
Furthermore, none of them have more people than Germany (~80 million), the country where the modern welfare state was born (Bismark put in place the first welfare reforms as a means to try to stop the growth of the socialist groups that were becoming a threat to the political stability of Prussia; though while he gave with one hand, he outlawed dozens of them with the other).
Of course, even the Scandinavian countries subdivides provisioning of these services further, down to the county level for many types of services.
In other words: This is a problem that decomposes very nicely, and that scales up accordingly. Just threat it as a suitable number of independent systems. They still have the economies of scale to make it reasonably efficient. You have some added complications, but also the benefit of being able to study differences in the various implementations and copy the most effective practices. Scale is opportunity here, first and foremost, not a problem.
> You also have to take into account the Scandinavian cultural norms as well.
As a Scandinavian, I'm very curious about what Scandinavian cultural norms you believe makes this easier that isn't present elsewhere. Besides, it's not just Scandinavia that has a comprehensive welfare system. Most Western-European countries do.
> As we've seen in recent years, when you institute measures like this it actually creates more dependency and self reliance on the government
This is only a "problem" when you don't consider that the alternative is that these people starve, live in substandard housing or go without important medical care. Nobody wants people to be dependent on welfare, but I prefer "welfare dependence" to having people grow up destitute.
If/when there are jobs that pay enough, fine, put strict conditions in place to push people to take them or lose benefits. But when there aren't enough jobs, or the ones that are available doesn't cover housing or food nothing good is achieved by not providing support for the basics.