Short Answer: This video summarizes the answer. Islam is valid for every time and place. If the prophet (peace be upon him) had appointed a successor, this would have limited succession to only one method, which is designation. He therefore left the options open for later generations.
Salam Dear Sara,
For your question, let us first start with your statement that many of the sahabah (companions) did not attend the funeral of Prophet Muhammad (peace be upon him).
If what you meant was that some of them did not attend his actual burial, that fact is very true.
In fact, the actual burial of Prophet Muhammad (peace be upon him) was attended by a few people, mainly members of his family.
The reason however, definitely had no relation with the thorough consultations over the issue of succession.
Why Didn’t The Sahabah Attend His Funeral?
At the death of the prophet, the companions found themselves facing the question of where to bury the prophet’s body.
There were two opinions: either to bury him in his mosque or to bury him with his companions in the Baqi`.
Then they found the solution in a hadith narrated by Abu Bakr, in which the prophet (peace be upon him) said:
Every prophet who died, was buried in the exact spot where he died.
Actually, the prophet (peace be upon him) died in `Aisha’s room, so he was buried there. Obviously, the room was not big enough to hold all his companions.
Thus, only the needed number, among the members of his family, carried out the burial procedure.
Such details, along with others, were what the editor meant by logistic reasons for the delay, stated in many sources and discussed earlier in my previous answer.
As for the janazah prayer, there is no evidence whatsoever, stating that any of the companions, who were in Madinah at the time of the prophet’s death, did not attend it, for whatever reason.
If you have any source, which states the contrary, I would appreciate your sharing it with me.
Why Didn’t He Designate A Successor Before His Death?
As, for the second half of your question – believe it or not – the prophet’s not designating a successor, is among the signs that this deen (religion) is perfect!
As this choice is a political one, it was left to the ummah to decide through shura.
Please do not forget that one of the major characteristics of Islam is its validity for every time and place.
If the prophet had appointed a successor, this would have limited succession to only one method, which is designation.
But, it would have also created another problem: how to determine who is better for the office?
He Left The Options Open For Later Generations
By not appointing a successor, he left the ummah with more than one option of how to select a caliph:
1) By letting the whole ummah choose the caliph – like what the prophet did.
2) By designating a successor – like what Abu Bakr did.
3) By appointing a council to choose a successor – like what `Umar did.
From these different choices, it is possible for the ummah to develop its political leadership selection system, in a way that would be suitable for the different social and political contexts.
Simultaneously, it would not contradict Shariah.
Imam Ali, at the last moments of his life, was asked by his followers to designate a successor.
If I designate a successor, there was a man who was better than me, who did actually designate a successor – Abu Bakr. And, if I do not designate a successor, there was a man who was better than me, who did not designate a successor – the prophet (peace be upon him).
I hope this clarifies the matter.
Thank you and please keep in touch.
(From Ask About Islam archives)