"But close to nothing is being recovered. We would recover more or less the same if we introduced a tax on bad weather to be paid by those suffering from it."

This is a good point, actually ...

If the tax is punitive enough that it results in a meaningful decrease in miles flown then it has a positive environmental impact, but it's difficult to imagine the taxes being that punitive. That's probably politically impossible.

In order for this to "recover" something, as is being discussed, you'd need a way to convert these dollars into carbon sinks of some kind - which isn't, itself, without controversy ...

I am personally in favor of removing subsidies to air travel and increasing taxes on miles flown such that the impact would be noticeable, but again, that's a tall order, politically.