Just to provide some numbers - the USAF's budget is $170 billion per year. For comparison, that's about the cost of Britain's entire NHS. A very large amount of that goes to airlift capability - plenty of money is spent on air superiority, but the USAF's primary mission in this era of asymmetric warfare is logistical support.
Basically, the attitude of most European governments is "If the US is willing to pay for it, why should we reject their help?"