Major airports and some routes between clusters of major airports, such as western Europe to the US east coast are already quite crowded with several at capacity.
Also, flight radar, especially at the zoomed out views, looks a lot more crowded than it really is, the airplane-icons are huge.
Crowding is a problem at airports, because each runway can only handle so many planes per hour.
Outside of that, the sky is a really big place. Not quite big enough that you can completely ignore collision avoidance, but big enough that you can pack in a lot more traffic than we currently do.
If you watch on a clear night from the roof of my building in Manhattan, you can see about 3-4 planes at a time lined up to land at Newark. Sometimes it seems like the Berlin airlift because they just keep coming.
One of the great things about the Berlin airlift is the way that they invented a huge amount of what is now standard high-capacity aircraft handling, on incredibly short notice (and in order to accomplish something fairly heroic). This includes things like standard routes and approaches, diversion procedures, pure-IFR (instrument) operations, and more. The family resemblance between arrivals to Newark Airport and Tempelhof is not accidental.
(Which reminds me, cycling down the runway at Tempelhof is still on my bucket list. I should go to Berlin this summer...)
You should. It’s very nice there during summer and there are a ton of pretty cool and informative info panels on the history of Tempelhof and the airlift. Bring a grill and a kite!
Also, flight radar, especially at the zoomed out views, looks a lot more crowded than it really is, the airplane-icons are huge.