Now we have twin engines that can at least do the Atlantic trip just as well, allowing all manner of smaller airlines to offer trips from a multitude of airports.
Technically they could but regulations didn't allow it at the time: 1953 FAA regulations required that twin-engined aircrafts always be within 60mn of an airport, and ICAO had slightly less restrictive limits of 90mn. This meant some routes were entirely unfeasible, or the flight path required to keep within 60~90mn of an airport at all time made it uneconomically unfeasible.
3- and 4-engined aircrafts were not affected by these rules (starting in '64 for the 3-engined ones).
The rules started being relaxed in the mid-80s with the introduction of ETOPS ratings (initially 120 and 180, now up to 370).
The 777 had already made the 747 largely obsolete, and the kings of long haul flights going forward will be Boeing 787-9 and Airbus A350-900. With probably a mix of both in most long haul fleets.
It's surprising that Airbus did hit the jackpot with its A330, seemingly by accident, and yet seems to have been unable to foresee that it really was the future. Boeing built a better A330, called 787, and even though the A350 is a success, simply by virtue of being a larger plane that goes further it will probably not sell as much as the 787.