Disagree a bit on this one. I don't think Advil is perfect in reducing a fever to zero and you have to time it right, etc. Second, I think the temperature taking acts as a deterrent for those people who would try to pull this stuff and fly sick. Finally, is there a correlation between your likelihood of transmitting the virus and whether you are symptomatic? Coughing alone seems like it would make it more likely to travel farther in the air. I think that Theresa Tam was pushing this a few months ago.
Where I do agree is that this is going to throw a tonne of false positives, and you'll get a lot of people who are going to be put out. That said, planes are different than the supermarket. These measures could theoretically prevent the introduction of the virus into new locations where it could spiral into millions of cases and tens of thousands of deaths, plus all the economic fallout, etc.. So I say bring it on and do whatever you can. The government is asking many people to sacrifice their employment while we get this under control. Telling people they can't fly with a fever seems like a pretty minor concession by comparison.
A Cough/Sneeze is definitely the way this spreads the furthest (on surfaces and in the air temporarily). A highly virulent person can still expel some droplets that contain some of the virus simply by breathing (and more of it when talking) even if they don't have any symptoms (yet). That exhale though contains a relatively small amount and doesn't spread that far. So if you're sitting next to (or having a conversation with) an asymptomatic but infected person for more than 10-15 minutes, the viral load you might receive could be high enough to give you the virus.
There have now been a few studies of super spreading events and it's almost always in an indoor environment where people are in relatively close contact for an extended period of time. Some of the ones I saw where a restaurant in China where one infected person ended up infecting a majority of the people that were "downwind" from him/her due to the restaurants ventilation. The act of eating and talking for 1-1.5hrs was spreading his infected droplets into the air, the draft would blow it to one side of the restaurant and most got sick. This virus is no joke.
COVID-19 Outbreak Associated with Air Conditioning in Restaurant, Guangzhou, China, 2020
From what I've read, to get the virus, you need a certain viral load to reach your throat/lungs. Different events cause the viral load to spread at different rates. A single cough or sneeze in your direction will likely be enough of the viral load to infect you but what's released while speaking normally would need about 10 or so minutes to reach a level that could potentially infect you.
Now if someone in public sneezes or coughs in your face, I think they deserve a swift punch to the gut at this point.
All this is the reason why physical distancing is so important followed by frequent hand washing. I suspect that with this virus though, the number of infections stemming from touching a surface is relatively low as the viral load would have to be high/fresh on that surface and you'd have to immediately touch your nose/mouth/eyes for it to enter your system.