Contrary to popular belief, quad engine planes are not inefficient [1], at least not in terms of fuel consumption (we are talking 1% to 2% difference). In terms of dispatch reliability however I think they are doing somewhat worse than twins, as they are more likely to have one engine out of service. And it not a small problem: a plane flying revenue passengers is an asset, but a plane not flying is a liability.

At least two of the planes you list are not being phased out because they were quads. The A340-300 was doing very well against the 777, however the A340-600 did very poorly against the 777-300ER. This is because the frame was too long and narrow, requiring extra reinforcement not to bend, thus ending up much heavier than the frame of the 777. The frame itself was grossly overweight, the extra engines were a rounding error.

The A380 was (and still is) extremely efficient (in terms of fuel burn per seat, at equivalent seating density), with a fuel consumption similar if not better than that of 787s or A350s, despite engines one generation older. It is, however, way way too big. In the end flying empty seats is very inefficient. But that has little to do with having 4 engines instead of 2.

The re-engined 777-9 will certainly be the most efficient plane flying when it is launched in 2021. Yet sales are lacking. It may, like the A380, turned out to be too big.

[1] https://leehamnews.com/2015/12/11/bjorns-corner-twins-or-qua...

The transition from 4 engines to 2 was accompanied by a great increase in systems reliability. Look up ETOPS for details. The trade off between safety (loss of airframe) and reliability( plane has all engines functioning ) is complex. A 2 engine aircraft is more reliable than a 4 engine aircraft, but a 4 engine aircraft can still fly safely with less than 4 engines. ETOPS made the design margins increase such that 2 engines would have identical safety and reliability.

One engine out situation is called for while planning the flight.

The need for all four engines being operable isn't changed by the viability of 3-engine flight. Once the plane lands its grounded for revenue service, and even three engine ferry flights are a huge affair to pull off.

So, de facto, each engine is strictly required, and having twice as many as a twin, quad jets are half as reliable as a twin, all other things being equal.

(me: licensed dispatcher who worked with the BAE-146 for a time. Incidentally, "BAe" stands for "Bring Another Engine" <grin>.)