I know of no multi-engined aeroplane in service on which an in-flight engine shutdown is prohibited. I also believe that there is no barrier to shutting down an engine as a precautionary measure and re-starting it at a later stage of the flight. Obviously, an engine which was shutdown in the event of an emergency, such as engine fire, uncontrollable propeller, etc would not be restarted unless some far greater emergency existed. My reciprocating multi-engine experience is limited to C47/DC-3 (P&W R1830) and Bristol Freighter (Bristol Hercules Sleeve Valve) aircraft. Multi-engine jets include Canberra bomber (RR Avon turbojet) and Meteor fighters (RR Derwent turbojet), turbo-prop multi-engine C130's (T56), Multi-engine TurboFan B707 (JT3D-3B) L1011 & B747 (Both RR RB211's). None of the above engines could not be shut down and restarted. In fact the T56 equipped P3 Orion often loiters on station on 3 engines, the engine being restarted when required. I think what is implied regarding "cavitation of fuel pumps" is that where an engine has been shutdown with the fuel supply shut off, and is unable to be feathered for whatever reason, may lead to secondary damage. The situation obviously would/could have far more implications than simply damaging an engine driven fuel pump. All the name calling too and fro is not enhancing the debate nor the standing of the combatants.
BTW, practice engine failure by shutting down the engine was/is discouraged due to the number of times such practice led to loss of aircraft and life, the events being far too numerous to list.