I'd say there is a big difference between people from an area using an airport, eg people of Leeds using manchester, as opposed to manchester fully serving the needs of Leeds. Eg where does most of the investment go? Manchester.
So, is manchester doing everything for Leeds? No, but people of Leeds use it be because they have no choice but the city doesn't benefit as much as it could if the services where available at Leeds.
Same true in London. Which is richer and more affluent? East or west London? Yes I know Canary Wharf and the financial district is in the east, but that works because of city Airport, and in fact it's what supports city (like oil in Aberdeen) Heathrow drives economic development along the m4 corridor, again people in East London use heathrow because they have to, but the investment goes around the M4 ... Or the cities with a direct air to Heathrow to give seamless trouble free(ish) journey.
The problem is, going back to Leeds, Leeds probably can't replicate the services manchester has alone so it's a viscous circle ...