I'd say the particular harm from it is if you take the claim literally to the point of assuming that there's nothing to learn from any other country. I'd also say that there's a problem if even the remotest way of objecting to the claim is taken as anti-whatever nation you're in. I am not sure, for example, that the UK can claim its response to Covid-19 has been the best in the world -- even if you claim that it was about the best we could have done under the circumstances, other countries have manifestly had better outcomes, and we shouldn't be afraid to say so and to attempt to understand why. It's not anti-British to be jealous of South Korea's success, both economically and in terms of their health outcomes (or Japan's, Finland's, etc etc.).
In short, if claims of being the best stop you from striving to be better still, or stop you from thinking there's anything to learn from other countries, that's a problem.