They say in the video they make about $8.90 per passenger (after taxes) over and above the $19.00 they need to break even. That's almost 50% profit. How is that bleak? That sounds like a good margin. Even smaller airports that don't make as much money directly from airport users would likely be helped by government due to all the collateral benefits and tax opportunities they could reap.
[EDIT:].... Ok I watched to the end of the video. They say 2/3rds of airports lose money. Again, government funding it is important for collateral benefits, employment of the local citizens which pay taxes, commercial benefits to local businesses, hotels, tourism, shipping hubs, trucking, etc. The money circles around and comes back from all of the other benefits, the government makes up for it and tax payers are willing to pay a tiny amount for the benefit of having that airport close by.
The largest airports privatize in UK because of mass-scale benefits. You need that because the margins are still not great, so having higher traffic, longer-haul flights, more profitable per passenger and many more passengers, etc... that helps increase profit. No small airport can run the same efficiency. Interesting video though.