Common sense is always a good idea, however one has to take care not to invent their own SOPs just because everything from boeing or the company is seen as normal corporate bull.
Anyway, to stay in the circuit you need a rather pressing issue in which case the non normal immediate landing (formerly called immediate return) checklist applies, which indeed does not require to clean up the aircraft. For a normal engine failure there is usually no time critical problem and it's best to just fly the standard procedure, do all the checklists, FORDEC, performance calculation (OEI landing climb limit has to be checked before departure in low vis to decide if you need a take off alternate or not), and then set up for an approach back.