If I am not mistaken there is a lift pump for the water to the engine. It is not the engine water pump, just a small pump used to lift the water to the engine. On some engines if the engine is started out of the water, it only takes a few seconds of dry operation, for the rubber impeller to melt and it stops pumping. Check a parts manual or operators manual for evidence of a lift pump. It explains your symtems perfectly. I had the inssue on an I/O years ago.
Hope this helps,
Kirk