From another POV:Halifax had no enemies (maybe,because he had no convictions? :he was an appeaser when the appeasers dominated,partisan of the war to the end,when Churchill was PM),he was trimming his sails according to the wind ,well,a politician .He was liked by Labour,he was the darling of the establishment.
and, Churchill:would it be an exageration to say that,before the war, he was detested by every one ?
And,Chamberlain :would it not be a good description to say that he was stubborn,with convictions ?
Whatever,about Halifax as PM during the war :I doubt that he would last ,because
1)while Chamberlain was keeping Winston on his place,I doubt that Halifax could do this
2)Chamberlain had the unconditional support of the Tories (majority in the Commons),but,it is not certain that this support would be inherited by Halifax:till the death of Chamberlain,Britain had in fact 2 PM's:Churchill and Chamberlain,but,after november 1940,the Chamberlainites became Churchillians,and, Halifax ? He was exiled to Washington(inimaginable while Chamberlain was there),and,no Tory protested .