Your first example is a classic food poisoning probability question. Assuming that 10% of eggs carry salmonella, what is the chance of NOT catching salmonella when eating one egg? It is 0.9 (90%). What about after eating 10 eggs? It is 0.9 per egg, or 0.9 raised to the 10th power, which is 0.35 (35%). So your chance of catching salmonella (or in your example, meeting a person with coronavirus) is the inverse, or 65%. That is not 10%.
We can also play this game with the airplane, using test sensitivity, test specificity, and disease prevalence to derive positive and negative predictive values for people who get tested, and then load them onto an airplane. You've already stated disease prevalence as 10% and test sensitivity as 90%. I'll add in test specificity as 95%. This is a classic calculation, source below.
Positive and negative predictive values - Wikipedia
The positive predictive value for testing under your given scenario is 66%, so that for every 2 true positive test results there will be 1 false positive, and somebody either booted from the plane or quarantined without coronavirus. That sucks, but until the test sensitivity and specificity improves, or the prevalence of the disease goes up, it's
But for the people who actually get on the plane, we are interested in the negative predictive value. This works out to about 90%, so about 1 in 10 negative tests are actually infected with coronavirus. What is the chance that 100 people with a negative test that get on the plane are actually negative? It will be the egg problem, but with 0.9 raised to the 100th power. That's a very, very, very small number (0.003%), so it's essentially guaranteed that there is at least one, and likely closer to 10 people on the plane with coronavirus, even though everybody tested negative.