Seems like that could be achieved with a static HTML website, possibly augmented with a REST API.
I doubt that information is changing very quickly. It could theoretically even be committed to a Git repo in a CSV file. However, although that would work for generating a website, it wouldn't be clone-able. So perhaps consider some canonical datastore in the cloud somewhere.
Also: I propose Apache/MIT licensing to minimize the friction of collaboration with both commercial entities and charities.
Technology wise, I don't care: it should just be something mainstream, popular and easy.
> Moreover, there's no way for anybody to know how many people have actually been tested, because the same person can be tested multiple times.
That seems like a harder problem because of patient privacy issues.