Regarding the destination information being up to date on airport websites, I don't think this makes much difference. I don't know of anybody who books flights by looking at the airport website and then checking. Most people either use a travel agent or a web search like skyscanner and then they will just search London-Budapest or wherever and find the cheapest fares. If someone did want to check where you could fly to from a certain airport, I would imagine many people would check Wikipedia. However, perhaps that's just the younger generation.
Even the best airports can have bad websites. The LCY route map on their site was never up to date. LCY also use to offer their own booking engine and again it always priced up flights at much higher than was available elsewhere but perhaps these book into different fare categories.