The key assumption you have made is assuming people in the US act sensibly, and that they would base their answer in the poll on facts. This assumption might not be accurate. A recent (Jan 2019) study shows that facts might not be too important in this matter .
Inaccurate views of scientific consensus and the willful rejection of scientific consensus.
In a 2014 US survey (15), two-thirds of respondents (67%) thought that scientists did “not have a clear understanding about the health effects of GM crops,” despite broad scientific consensus on the topic (16). Similarly, half of Americans (52%) thought scientists were “divided” in the belief that the universe was created in the Big Bang, and about a third each thought that scientists were divided on anthropogenic climate change (37%) and evolution (29%). Of course, these data do not make clear the cause of these inaccurate views, which, arguably, could stem from people being uninformed, intentionally misinformed, or a bit of both.
Furthermore, split-ballot survey experiments have shown that even when Americans do seem to possess accurate knowledge of scientific consensus (however large or small that number may be for a given issue) there is no guarantee that they will integrate that knowledge into their attitudes or policy preferences (17). In other words, these respondents know what the scientific community has established as fact, but they nonetheless refuse to “know” it. Some have therefore argued that rejection of scientifically accurate accounts of the Big Bang or evolution by nonexpert audiences indicates neither a lack of information about scientific consensus nor the presence of misinformation, but, rather, motivated information processing (18). (Highlight mine).
Given that the current US government places little value on facts and that "Make America Great Again" was a slogan in the last election, I find it highly plausible that this is a factor here, too - that a certain amount of people simply want to believe that the US is the best in the world in everything, despite facts showing otherwise.
The study also finds:
When such directional goals influence reasoning processes, individuals are prone to “biased assimilation,” which is characterized by confirmation and disconfirmation bias, or the parallel tendencies to privilege information that is consistent with one’s predispositions and to discredit information that seems contradictory (51). As with selective exposure, motivated reasoning can contribute to an individual becoming misinformed, and it can occur not only in political contexts but also when individuals process information about science and emerging technologies (52–54).
If you want to believe the US is #1 in everything, this seems to be a highly relevant point.
Also, emotional state can be an important factor. And I think we can agree that the current crisis is a very emotional matter, and that believing the US is doing better than anyone else might be reassuring, even if it isn't based on truth.
The role of emotion.
This brings us to discussions of the influence of affect in motivated reasoning processes. There is some evidence that a person’s emotional state can shape the accuracy of his or her beliefs. [...] Notably, individuals’ attraction to emotionally charged content is not limited to politics, and even when it comes to scientific discoveries, individuals are more inclined to spread information that has a greater emotional impact (59).
1: Scheufele, Dietram A, and Nicole M Krause. “Science audiences, misinformation, and fake news.” Proceedings of the National Academy of Sciences of the United States of America vol. 116,16 (2019): 7662-7669. doi:10.1073/pnas.1805871115
So I think the answer is not that there are any objective metrics that show that the US is handling the crisis better in any way. The fatality rate is magnitudes higher (although it is too early to tell what the toll will be in the end), cases per capita are higher, total cases are higher, et cetera. I don't think there are those metrics that you are looking for. I rather think think it simply shows that a large percentage of the US population is either uninformed or misinformed, for various reasons as outlined above.
From the survey that was linked in the question, we can also see that the opinion on how the US is handling it relative to the rest of the world varies dramatically with the political orientation of the respondent, which would further back up that political orientation plays a significant role in how people perceive facts and responded.
The poll data for the question relative to South Korea is roughly in the same boat, polarization-wise. Only about 6% of Republicans say the US is doing worse than South Korea, whereas 45% of Democrats say that: