By about 1981-82, with the 757/767 in the offing, the push was on for ETOPS approval "out of the box", on initial delivery of new aircraft.
Not quite - the 757/767 helped pave the way for EROPS/ETOPS, but they didn't have it out of the box, nor was there much consideration of ETOPS during the design phase. During the early transition to what was initially called EROPS, the FAA required a significant amount of in-service time on the engine and an established (and good) in-flight shutdown rate (IIRC 250,000 hours before you could even petition for 90 minutes). It wasn't until the engine upgrades to the 767 in the late 1980s (PW4000/CF6-80C2/RB211-524) that the real push for 180 minutes came into play. I recall one big meeting where management explained that the 767 now had enough range to fly from Seattle to New York and back without stopping, but since there was little demand for that we needed to get 180 minute ETOPS so that the operators could put that range to good use.
The 777 was the first aircraft designed from the ground up for ETOPS 'out of the box'. I was on the program at the time and we were quite literally making it up as we went. The processes we came up with during the initial 777 development have been largely adopted as the requirements for early ETOPS.
To answer the original post - two engines are more economical for several reasons. First, two engines require less maintenance that three or four. Engines tend to be a bit 'draggy' as they interfere with the airflow around the wing - less is better. Engines and the associated nacelle and hardware are heavy and expensive - two big engines tend to weigh and cost less than three or four smaller engines.
But the biggie is that, due to the differing TO vs. Cruise thrust requirements between two and four engines, twins tend to operate in a more favorable area of the "TSFC bucket" at cruise relative to quads. Now, you could put more powerful engines on the quad to get that same TSFC bucket relationship, but those bigger engines weigh and cost more and have more drag that what's needed for TO performance, so it still ends up costing more.