Sounds like any Radial engine. Aircraft have short, essentially open exhaust (which you could not run on the street with any engine) and manual (pilot operated) mixture control.
On your example in the video he's running full rich fuel mixture; a pilot will only lean the mixture when he's climbed to cruise altitude. A racer like the example would probably run slightly rich the entire flight, as too much fuel helps with cooling.
Cerebration or fuel injection on aircraft are optimized for cruise RPM (around 1800). Radial engines are often named for displacement in cubic inches, and can be more than 100 cu in per cylinder. One of the most popular P+W radials is the 1340 series (1344 in2) = 150 in2/cylinder. So poor atomization at idle also plays a role.
This particular aircraft uses a P+W Double Wasp; two-rows of 9 cylinders (total 18) displacing 2800 in2
So, open exhaust and quite a bit too much fuel, combined with the radial's natural somewhat irregular misfiring idle.
The smoke when a radial starts up is because oil leaks past the rings and valves on shutdown, partly filling the bottom cylinders with what amounts to an oil + fuel mixture.
Most radials to my ears sound like a small group of idling Harley-Davidson twins, which is no surprise since the 45-degree air-cooled V-Twin is derived from radial engines.