Khilafah/HT Questions 2 - The Fard of working for the Khilafah

The Fard of working for the Khilafah

The Hizb refers to the Hadith, “Whosoever dies without having a bay’ah (pledge of allegiance to the Khaleefah) upon his neck, dies a death of jahiliyyah.” as evidence that it is fard to elect a Khilafah. Given that the Khilafah system (which would allow Muslims to hold elections) is absent, the fard to re-establish the Khilafah becomes an obligation through the fiqh principle of, “whatever is necessary to accomplish a wajib is itself a wajib”.

The time limit to elect a Khalifah according to the Hizb is 3 days and 2 nights and from what I read they do not use the events after the death of RasulAllah (saw) and the election of Abu Bakr (ra) as evidence. Rather, they use the events prior to Umar (ra)’s death and his instructions on electing the next Khilafah as evidence citing it as being ijmah as-Sahabah. The following is a
section from the book Ajhizah:

As for the fact that the maximum time limit allowed for the Muslims to appoint a Khalifah is three days including their nights, its evidence is that when ‘Umar felt that his death was imminent he delegated the people of the Shura to appoint a Khalifah giving them a time limit of three days and instructing them to kill anyone who disagreed with the group once the three days had lapsed. He assigned the execution of such instruction, i.e. killing the one who might disagree, to fifty people from the Muslims despite the fact that the group was formed of the Shura people and the senior Sahabah. This took place in the presence of the Sahabah and no one objected or condemned such instruction. This became a general consensus of the Sahabah stating that it is forbidden for Muslims to remain without a Khalifah for more than three days including their nights and the consensus of the Sahabah is Sharia evidence just like the Qur’an and the Sunnah.

In summary, the post of Khalifah has been vacant for more than the time limit (2 nights and 3 days) and the system (i.e., Khilafah) does not exist and with the fard not being fulfilled the duty expands to encompass all Muslims in the Muslim world or non-Muslim world.

How is this understanding incorrect?