How can one respond to those who claim that the Messenger (peace be upon him) died without bequeathing the caliphate, citing the hadith: "Let not one of you sleep until his will is under his head"?

Bequesting is recommended. As stated in the Hadith: "It is not befitting for a Muslim who has something to bequeath to spend three nights without having his will written down." Bequeathing may become obligatory if a person owes a debt, has a right against someone, or holds a trust, and so on. The meaning of the Hadith is that it is prudent and cautious for a Muslim to have his will written down with him. The Prophet, peace and blessings be upon him, did not write a will for his Companions regarding the matter of succession due to a wisdom willed by Allah.