IMHO: Lord Halifax it would seem did not want the burden of the office of PM at that time. In short, he was scared to assume that role.
Especially since he thought that Britain would loose to the Nazis.
He wanted Churchill to take that horror on, and the blame. Then Halifax could rise and assume command of the Nazi vassal that was the UK. Sieg Heil!
Some think he might even had had a secret channel to the Nazi’s planning to do so.