Government has to run, it has to work so until there is a new leader they have to work with him.
What do they do until a new leader is appointed ? Refuse to work with him and basically the country has no government.
I don't agree with it and don't fully understand it, unless you know better.
Firstly, there is a deputy PM, they are there for precisely such an occasion as this. He could resign today, Raab is immediately temporary PM and a new government installed, likely with many posts re-appointed as is, and it could all be done within a day or two.
Seondly, Boris can say "I resign effective October 1st" and the Tory MPs can say "Nope, no-one will accept any government positions under you, we will now go to the 1922 committee, change the rules and trigger a VoNC, at which point you're out early next week".
Thirdly, a decent sized group of MPs could make it clear they will not vote for anyone during the leadership elections who consents to this temporary retainment of power.
Imagine the damage to the Conservatives: dozens of MPs including government officials and secretaries of state have come out and publically stated Boris has failed to meet the standards of a PM and of being a serial liar. How can any temporary government have any authority or credibility in such a situation? How can he represent the UK internationally under such circumstances? Every single debate and PMQs they will get eviscerated because they are being lead by such a person. Nothing the government states between now and that date would be credible.
What would be concerning is if he does make this statement because it would almost certainly have been cleared with key sections of the party, he wouldn't come out and say this if he didn't know it would be accepted.