Mark - the logic of the statement that having a "recent" test means that the risk of contracting the virus in the short time afterward is very low - is that few people are exposed to the virus, especially those engaging in responsible practices (like wearing masks, avoiding large groups, social distancing, washing hands, etc...) in any given set amount of time. Of course the statement is completely dependent on what "recent" means, which is a valid question. The longer the duration after testing, the more chances there are of someone being exposed. It is also not clear how long it takes for the virus to be detected in one's system in a test. In your anecdote, someone might be sneezed on walking into the testing clinic, then test negative because the virus had not taken root yet. We have to be reasonable here.
I think a test a day or two in advance of the flight would be a reasonable condition, and make it easier to arrive comfortable that you would not be turned back because of a positive test if you wait to be tested on arrival. The challenge is that sort of on-demand testing may not be available in the US by mid-June.