The point about trying to lead the government from the House of Lords is a good one.
It technically wasn't impossible, which is why Halifax was even considered in the first place. However, since the Parliament Act passed in 1911, supply (money for the government) was voted exclusively by the Commons, and governments were made and unmade in the Commons. The leader of the Commons would have had so much authority in the government that a peer as Prime Minister would have effectively been a figurehead.
Even before the Parliament Act, in 1905 Campbell-Bannerman rejected a scheme for him to take a peerage -his health was bad and he died a couple of years later- precisely because he knew it was a scheme for him to be a figurehead PM. In 1916 Asquith resigned instead of submitting to a complicated scheme that would have reduced his powers by taking away the day to day management of the war from him. Asquith's health was also bad -his son had died a few months before- so the scheme would have made some sense but Asquith had been one of the proponents of the 1905 plan and knew it would turn the PM into a figurehead. By the way, the 1916 situation is an interesting "what if" that you don't here about these.
And the leader of the Commons in 1940 would have been Churchill, whose support was needed at the time by any government, since he had been proven right over appeasement (just as this year Boris Johnson could be blocked from becoming PM but had to be included in the government due to being on the winning side with Brexit). Churchill at the least would have been Secretary of State for Defense. So the choice was really between putting Churchill plainly in charge, or putting Churchill in charge with Halifax as the public face with a sort of veto over what Churchill did. Halifax more or less put it this way in his memoirs, without spelling the whole thing out like I have.
A peer did become Prime Minister in 1963, but by that time provision had been made for peers to renounce their peerages and run for a House of Commons seat, and that is what happened in that situation. Now they could have provided for peerages to be renounced in 1940 but this apparently not considered. In another discussion on this board, someone posted that the idea of just allowing members of the House of Lords to sit and speak in the Commons had been considered in 1940. The reasons why this course wasn't adopted wasn't given.
Now Halifax as a sort of figurehead PM, with Churchill effectively running the war effort as Defense minister does have some implications. Probably Halifax steps down mid-way and Churchill becomes PM anyway. Halifax may even wind up as Ambassador to the USA as IOTL. But this means Churchill is not PM in 1940-1, when he was most effective. OTOH, the arrangement may have alleviated some of the decline in Churchill's health that occurred over the course of the war.