Yeah, I get that it is the common approach but it doesn't stop it being quite bizarre when you think about it. And given the situation we are in makes it even worse.
I can't see what the alternative would be - you need to have a PM as there are things that require a PM all the time (not least the visit of Trump in a couple of weeks time).
However whichever way this could be sorted (continue as PM until successor is found, or drop in deputy as interim) the nominal PM will have no authority to act in any meaningful political manner until the new leader is found.
I think the question now is whether the Tory parliamentary party hold their nerve and put forward tow candidates to their membership, or blink and settle on a single successor (as per May and Brown) without reference to the membership. If the latter then a new PM could be in place by mid June - if the former it might not be completed until September, allowing time for hustings, debates and voting by the membership.