Why no more three holers?
Turn the question 180 degrees: Why were the three holers designed 30-35 years ago?
And the answer: Because the rules were different at that time. You had to have minimum three engines to fly any substantial distance where there were not plenty of airports underneath. With ETOPS rules of today the situation has changed completely.
The market for four holers more and more looks like a niche market. We saw ETOPS 60, then 90, 120, 180 and even 180+15% (207) and of course 240 is the next thing to discuss.
Forget about what passengers prefer. Most passengers don't even remember the number of engines on the plane they flew on five minutes ago. Airlines and airliner manufacturers decide the number of engines with respect to the rules drawn by FAA and other CAAs.
Hopefully we won't see ETOPS related incidents or what is even worse. Should that happen one day, then the rules may very well be rolled backwards again. It will increase the demand for four holers.
The A330/340 design seems to be a very clever move. It lets the customer decide the number of engines he wants on basically otherwise the same plane. And there are still Pacific or polar routes which are best served with four holers with present ETOPS rules.
The rest of the four holers are more complicated to explain. The 747 is a 35 years old design, which has come to a crossroad of its further development. It may be developed into something bigger, or it may fade away slowly. We don't know yet.
The last one A3XX is (also) too big to be powered by less than four known engines. But then we don't know if it will ever fly. The only thing we know pretty sure is, that if both an enlarged 747 and an A3XX goes into production, then at least one of them - maybe both - will have very hard times and will probably very soon fade away.
Then of course there is the niche bird above all other niche birds, the Avro RJ-70/85/100. It's a different story - not made for maximum economy, but made for going places where no other similar jet bird goes. It reduces needed runway length, not by noisy power, but by substantially reducing the spare runway the twin needs in case of engine failure at V1 speed.
Best regards, Preben Norholm
Always keep your number of landings equal to your number of take-offs