Most of the cases in New York are concentrated in and around NYC, which is an international travel hub with 3 very busy international airports in the area. I don't think there are many places in the world which are going to have had the same number of asymptomatic and symptomatic carriers pass through. So on that basis I don't think it's a good benchmark for most other cities or countries to use.
I wonder if it's better to compare Swedens overall per-capita infection rates with other states in the US where there is a lockdown but whose capitals/major cities are more secondary cities from the perspective of international travel. Let's try North Carolina (picked for being closest in population to SE and also being not a top airport hub or travel destination compared to its neighbors) though it does have significant business and educational centers, it's probably a fair comparison to Sweden wrt population density.
NC total population: 10.5 million
NC confirmed cases: 7,285
SE total population: 10.2 million
SE confirmed cases: 11,445
So anyway, there are other states and countries that are doing better and worse than Sweden, but I don't think that Sweden is in any way demonstrating that it is doing better by trying to ignore the problem and possibly to blame the lack of strict enough sanitation ("lack of supplies and poor routines"), it's just coming later than to higher-volume travel centers like WA, NYC, the UK, France, Italy, etc.
In any case, the idea of doing better or worse will, in the end, be measured in per capita deaths at the end of this disaster, and we haven't gotten there yet. I truly hope that Sweden finds its way through with as few deaths as possible, but I don't think that's what they've done.