I learned a long time ago, Karl, not to rely upon airport websites for lists of destinations served. US airports are the worst of all, either not having a list of destinations or including destinations which aren't served directly, but most other airport websites fail to mention routes which haven't yet come into operation (or show routes which are due to end in the near future with no indication that they are). Wikipedia tends to be far more reliable; if an airline announces that they'll commence a new route in 6 months time the relevant Wikipedia pages are usually updated within an hour or two, whereas the airport web pages might not get updated until the new route actually comes into operation. (I don't solely rely on Wikipedia though; I cross-check my information with the airlines' own websites).
237SJ's suggestion of using Lufthansa sounds expensive to me (but it might be worth checking out anyway).
Adding in Lisbon as an alternative starting point makes things a lot easier. Wizz can get you to KEF via Budapest, Katowice, Vienna or Warsaw Chopin. (Ryanair also operate between Lisbon and Budapest so, if the Wizz flights don't connect well, that could be another option for that sector. Similarly Ryanair-owned Lauda fly between Lisbon and Vienna).