Three hour ETOPs is not something that you just begin to fly. The authority takes years to get. It requires special aircraft certification, an ETOPs approved maintenance program, crew certification, and a reliabiliity history.
Often ETOPS planes will have additional or different equipment on them. Reliability of certain components is watched very closely and detailed records are kept.
In order to get two hour ETOPS you must fly 90 minute ETOPS for a while. Engine and APU reliability data is collected. Once you do 90 minute ETOPS for a while you can begin 2 hour ETOPS. Again, data is collected. If your reliability is not sustained you will lose the 2 hour ETOPS. That happened to Pan Am around 1990 using the Pratt engines on the Airbus 310s. That increased their fuel costs by 40 million per year on the ETOPS routes.
Once you have successfully done the 2 hour ETOPS you can go for the 3 hour ETOPS. The entire process takes years. It is more of an airline as a whole procedure than an airplane specific certification. If you have been flying 767 three hour ETOPS you can get 757 or 737 three hour ETOPS in less than a year. The FAA would have already been monitoring your ETOPS maintenance program.
America West did not have any 2 hour ETOPS routes to build into the 3 hour ETOPS. That is why they were never able to get it on their own. They went to the FAA and tried to get them to agree to a simulated 2 hour program going cross country. The FAA told them that if they had a problem they could not overfly good airports to fly to their simulated ETOPS airports.
Everytime an ETOPS component breaks the data goes into the mix and you are a little closer to losing that certification. The smaller the ETOPS fleet the greater the component failure figures into the equation. The larger the ETOPS fleet, the more components that can break. And there had better be no paperwork errors, especially for the three hour ETOPS.
Basically, it is an extended process. Some of this information may be a little out of date since it has been a few years since I helped on setting up an ETOPS program.