I agree that this sentiment is a major issue for public support of welfare systems. A few (admittedly speculative) thoughts on what's different in Denmark, where such sentiment isn't absent, but I think is a bit weaker and less influential:

1. In a way, the U.S. makes for a foil that enhances the perception that the welfare state is necessary for a society we'd like to live in. People are afraid of Copenhagen ending up like San Francisco or Detroit, with homeless people everywhere, dangerous ghettos, etc., and there's a perception that the welfare system is one thing keeping that possibility at bay, and keeping Copenhagen a fairly safe and nice place to live. The "are you trying to turn us into America?" trump-card is a huge PR/positioning problem for the (smallish) free-market/libertarian parties, because a lot of people are genuinely afraid of it.

2. A decent amount of stuff that is "welfare" in the U.S. is instead just something everyone gets, so there is no issue over eligibility and whether people are gaming it. Everyone gets free healthcare, subsidized childcare, free university, paid paternity/maternity leave, etc., so there aren't debates over things like Medicaid or CHIP. Even the public-housing system is open to anyone (there are waitlists for popular buildings, but no income/wealth cutoff).

3. There's a general feeling that even people who really are up to no good are still somehow "our responsibility". It's kind of the society-level version of what southern Europeans feel on the extended-family level. To a Greek, your uncle might be a lazy jerk and/or an alcoholic, but he's still your uncle and it's seen as at least partly the family's responsibility to make sure he's not homeless or resorting to crime to eat: it'd be shameful to the family if one of theirs was not taken care of. In Denmark it's sort of the same, except it's not the individual family's responsibility, but the state's, to make sure everyone in the "extended family" is off the streets and able to eat.

4. Low-end jobs just pay a lot more. Even working as a supermarket check-out clerk, barista, or McDonald's employee, pays ~$35k-$40k if you do it full-time. So being employed makes a qualitative class difference: in the U.S., minimum-wage workers and welfare recipients are both "lower-class" or "poor", whereas in Denmark, welfare recipients are mostly "poor" while minimum-wage workers are "middle class". As a result, people tend not to feel envious of those on welfare, but more pity towards them.

5. There's a greater cultural acceptance of the idea that people can be temporarily disabled. You can be on disability or sick leave for 6 months for a physical or mental health problem, and then go back to work; it's not only for people who are permanently disabled. This is seen as relatively normal, and such sick leaves are relatively common. Since many people have either taken one themselves, or have friends/family who have, it's not seen as much as something that "other" people are doing to take your tax money, but something that many people, including your own social circle, might need.

In general, the fact that many people will use the welfare system at some point in their life, even if only briefly, means people are very interested in ensuring that it's quick and efficient. Whereas my impression is that most Americans don't have a personal interest in the welfare system being quick and friendly, because they don't expect to use it.