If you read the CVR transcript,
https://data.ntsb.gov/Docket/Documen...elease-Rel.pdf
the most important item, validation the faulty engine, gets the least amount of attention. They spent 4 minutes on RT after identifying the true defective engine and after the RT they stick to the wrong one.
On a jet you don't face much of drag problem of a faulty engine, so no need to shut down anything quick unless on fire. That part they did right. And before ditching try anything which can increase the noise level.
Simple with 20-20 hindsight from the armchair. The question remains, why did they not go back into problem identifying when the assumed good engine could not keep altitude?
They had 4 minutes from realizing that to ditching. 141.10 to 145.17