All of the big three had, and still have the ability to build engines that are reliable and produced that level of thrust.
GE was chosen in a large part because they offered Boeing a deal they could not refuse. In short they said: Give us engine exclusivity and we’ll give you a large payment to go towards developing the airframe. Who would refuse that.
Only having one engine type keeps development costs down as no need to design multiple pylons, nacelles, fuelling and cooling systems.
Also it allowed GE to optimise the GE90 and spend more on it, as they knew Rolls Royce and P&W would not be taking away market share.
Rolls Royce did a very similar thing with the A350.