merc did make some alpha packages with closed cooling and found out rather quickly the sea water pump in the drive could not provide ample cooling water for the heat exchanger/engine, hence the engine mounted "bravo" style sea water pump
the ones i've seen have the through transom water passage hose from the drive cut on the outside and the through transom barb removed and capped off, this is so the pump in the drive still provides cooling for the drive but does not enter the engine compartment, all heat exchanger cooling water is provided by the engine mounted sea water pump. this is a very convoluted way of setting up a sterndrive cooling system which is why there aren't a whole lot of these packages out there
to op: if you are running your engine on the trailer with muffs you need to provide two sources of water to the package, one on the through hull pickup for the engine mounted sea water pump and one on the drive for the sea water pump in it, the way your pics show you have no water on the drive and if you've run the engine that way the drive pump is shot, also as stated keep the drive in the down/in position whenever you run the engine. best to change both pumps and housings including the base assy for the drive sea water pump, even though the impeller(s) look okay often times they will have a set that does not allow proper volume for cooling