I know that there is a considerable difference between the UK and the US in airshow safety regulations, and they have only got worse since the Shoreham Hunter crash. Our previous airshow crash with fatalities on the ground was when John Derry's DH110 Sea Vixen prototype crashed at Farnborough in I think it was 1955. Back then in the British traditional way, they tended to the casualties, cleaned up the mess and after a couple of hours delay went on with the show! Still it did result in the UK adopting some of the most stringent safety rules in the world.
So bottom line is the minimum focal length I would recommend to anyone who wanted to take airshow photography seriously would be 400mm, and I would strongly suggest a zoom. This pretty much leaves you starting with the 100-400 L V1. I used to rent this lens regularly for shows, and even when using it on a 300D it makes a huge difference in quality between that and any of the xx-300 consumer zooms. I really highly recommend the Sigma 15-600 C. It is optically at least as good as all of the 100-400's I have rented, and that is comparing them at the longest focal lengths, so 400mm against the sigma at 600mm.
Also here in the UK the Sigma C is a very good deal price wise. I just checked B+H's price, and I see that it is $989. They also show prices in Pounds Sterling, so I am basing my comparison on that. Here the lens is only £699 new, and that includes our 20% VAT, which has to be included on all displayed prices. If I remove that VAT and convert using the B+H rate it makes that lens only $726.96, even with the tax it is only $872.35, so it seems things can actually be cheaper in the UK. Still if you can find a Sigma C at a good price second hand that would be my recommendation.
From what I have read the 55-250 optically is the best of the bunch when looking at consumer 70-300's, even after cropping to match FoV. Still I would say that a good second hand 100-400 V1 would still be my minimum choice for a purchase.
One of the big reasons that I recommend such long focal lengths is that they offer far more scope for nice images. It is all very well getting them close enough that you can just get away with a 300mm lens when they are side on at closest approach. If you want to go with a 3/4 view then all of a sudden the aircraft become a lot further away. The longer lenses give you far more chances of an interesting shot that will come close to filling the frame.