My thoughts, having flown with them for awhile.
Yes, bishops have larger numbers than everyone else. This helps them maintain a stable 2-front war, but does not give them the success that they have had. Indeed, there were times when bishops outnumbered both knights and rooks by a large amount, and still lost bases on a consistent level.
bishops I have noticed, do one thing more consistently than any other country. They participate in Missions. Flying against bishops, I have noticed time and time again the standard tiffie raids that they produce. Their missions are simple, and easy to operate. They have individuals that have been around AH for a long time, and have flown bishops for most of that, so everyone knows them, and will follow them in a mission.
Knights used to do this aswell, and Ripsnort still does do some good missions, however, I think Ripsnort has been grounded in RL much more recently. (not sure) I haven't noticed as many missions coming from Knightland lately, although, I too haven't been flying asmuch as I would like.
People would argue that one of the reasons bishops missions are a success is because they have more people, therefore more will join. This is not the case. The mission simply has to be against the right target at the right time. I once (a long long time ago) ran a successful lanc raid against both the City (when there was only one) and the Radar installation of a country, coordinating them both at the same time. We had 8 lancs, no fighter cover, and just alot of good timing.
Anyway, to make a long story short, MISSIONS are a primary key to taking bases, and resetting an enemy. Make missions, participate in them, and quite possibly, you too could win.