Just my opinion and experience with the Sigma 150-500 on a crop body. Don't get me wrong, it's a fine lens and capable of great shots but... Racked out to the 500mm end and wide open, it's not as razor sharp as it could be, especially if you pixel peep in the corners. Stop it down a bit (f/8 should be far enough) and things improve. At the long end, it's not easy to track flying objects - the frame isn't huge and it can be hard picking a plane ou from the sky! Focus lock on isn't as speedy as it could be either (at the long end, on a fast moving object wandering around the frame!) but once it has locked on, it follows and tracks well.
Last year and the year before, I shot a few planes at the Dawlish airshow - more a glorified flypast really - and used my 70-300 VR Nikkor in preference to the 150-500 (which I dragged up the hill and didn't use!) The biggest benefit of the 70-300 is the light weight compared to the 150-500, although I negated that by dragging the heavy one as well!
IF your technique is good enough, the 150-500 will give you the extra reach (especially on a crop body) but the 70-300 may well be enough, especially if the MP count on your body is high enough to allow some cropping.
Lancaster with Dawlish, Dawlish Warren and Exmouth as a backdrop. f/20, 1/100th s, ISO 200 at 300mm on 70-300 VR Nikkor on a D700.
Vulcan with Red Arrows as escorts. f/10, 1/500th s, ISO 200 at 155mm (232mm 35mm equivalent) on a D70.