The OAA statistics represent the first job students obtain after leaving employment, not every job they get in their entire career.
Many major airlines run cadet schemes, or have a history of doing so, prime example being British Airways who will without doubt re-start this tradition at some point in the future. The likes of Qatar are starting to run such schemes through flight schools. In terms of it being a 'major' airline, well in terms of fleet size and customers, easyJet and Ryanair are major airlines even though they are low-cost, they take cadets and the flying is no different, operating the same a/c types as you would at BA or Qatar.
If you go through OAA, depending on the criteria of the airlines you are able to apply for the cadet schemes as they become available. The Cathay Pacific scheme I believe is not run through OAA and is a separate application.