Every time an engine is shut down in flight, it is tracked.
Engine reliability does affect ETOPS certification. In addition to the aircraft being certified, each airline is also ETOPS certified. An aircraft or an airline with a poor record could have its ETOPS certification downgraded, or even potentially revoked.
Wikipedia ETOPS Approval
ETOPS approval is a two-step process. First, the airframe and engine combination must satisfy the basic ETOPS requirements during its type certification. This is called "ETOPS type approval". Such tests may include shutting down an engine and flying the remaining engine during the complete diversion time. Often such tests are performed in the middle of the ocean. It must be demonstrated that, during the diversion flight, the flight crew is not unduly burdened by extra workload due to the lost engine and that the probability of the remaining engine failing is extremely remote. For example, if an aircraft is rated for ETOPS-180, it means that it should be able to fly with full load and just one engine for 3 hours.
Second, an operator who conducts ETOPS flights must satisfy their own
country's aviation regulators about their ability to conduct ETOPS
flights. This is called "ETOPS operational certification" and involves
compliance with additional special engineering and flight crew
procedures in addition to the normal engineering and flight
procedures. Pilots and engineering staff must be qualified and trained
for ETOPS. An airline with extensive experience operating long
distance flights may be awarded ETOPS operational approval
immediately, others may need to demonstrate ability through a series
of ETOPS proving flights.
Regulators closely watch the ETOPS performance of both type
certificate holders and their affiliated airlines. Any technical
incidents during an ETOPS flight must be recorded. From the data
collected, the reliability of the particular airframe-engine
combination is measured and statistics published. The figures must be
within limits of type certifications. Of course, the figures required
for ETOPS-180 will always be more stringent than ETOPS-120.
Unsatisfactory figures would lead to a downgrade, or worse, suspension
of ETOPS capabilities either for the type certificate holder or the