I would not be too worried about head spacing or accuracy,
SCmackey, but there is no escaping the fact that the spring-loaded detents that secure the belted rimless cases in the ejectors or extractors are more complex and delicate, hence more prone to failure, than ejectors or extractors designed for rimmed cases. It is just mechanics 101.
It is a fact that they do work most of the times, and many users may never experience any failure to extract / eject, but it is also a fact that they sometimes fail, even in the highest quality rifles. How lucky do you feel?
A good alternative to the .375 H&H in a double, if the purpose is to get more ballistic flexibility than the .470 or .500 offer, is to consider the modern 500/.416 which duplicates the performance of the .416 Rigby.
In the end, because the .375 should NOT be used as a "stopper" caliber (it lacks power for the purpose), hence the consequence of a failure to extract / eject is likely just annoyance -- as opposed to failure to reload in time to stop a charge; and because a client user is backed by a PH with more serious artillery when hunting DG, I would not necessarily pass on a .375 H&H double, but this would be with the specific understanding that I do not rely on it as a "stopper".
EDIT:
as I read Canadaboy's input after posting my response (I was typing when he posted), I observe a similarity of concepts... My own experience with doubles was with 9.3x74 in Europe, then .450 #2 and .470, but I am off the doubles bandwagon (recently resold my Krieghoff .470) and have become a Blaser R8 only gunner. My second shot (whether it be with my .375 H&H or .458 Lott barrel) is very darn near as fast, as I am reloaded by the time I come down from recoil, and my third and fourth shots are light years faster... Beside, I gained infinitely more R8 practice in just a few years than I did in 30 years of doubles ownership...