TC, that is the most plausible hypothesis I have seen so far.
The pilot was not worried about fuel, and accepted additional non urgent tasking. His gauges told him (erroneously) he had sufficient fuel, and when the red warnings appeared, they still showed sufficient, and there had been no amber warnings. He concludes that there is a fault, rather than a fuel shortage, and as he is enroute to base anyway, puts it on auto as you suggest and begins to troubleshoot.
When fuel starvation affects the first engine, still convinced by his gauges which are showing (erroneously) sufficient fuel in the supply tanks, he switches on the prime pumps as part of his response to what he percieves is a problem in getting fuel from supply tanks with plenty fuel, to the engines. Convinced the problem is not fuel quantity, a view supported by his contents gauges, he continues to head for the ever closer home base, which represents by far the best option for landing, and where the problem can be readily investigated and resolved without becoming front page news as a result of a precautionary powered landing within spitting distance of base. Alas we know the rest.
Assumes that the fuel probe problem was present , but if it was, it could explain why the red warnings, and even the first flameout were not properly diagnosed by such an experienced pilot.