Prior to the adoption of the the 25th amendment in 1967 there was no mechanism to allow a mid-term VP replacement. Presidents and VPs were to be elected only during regular years. If a president had to be replaced the just went down the list to find the next appropriate successor. So when Johnson replaced Kennedy they did what they had always done, and muddled along without a VP for the remainder of the term.

Senator Birch Bayh in his book One Heartbeat Away: President Disability and Succession specifically attributes the passage of the 25th amendment to the Kennedy assassination. Apparently seeing Speaker John McCormack (74 year old) and Senate President Pro Tem Carl Hayden (86) sitting behind Johnson at a joint session of congress as his nominal successors shook the lawmakers. Either would have been the oldest president ever.

One can argue that they got it done just in time as Ford was the first VP to be appointed according to the new 25th amendment, just six years after it went into effect.

As a follow up, was there ever any discussion of whether the Vice President could be considered a position that the President could appoint (if vacant) similar to other offices, e.g., Attorney General?

The scenario would be: President dies, Vice President becomes President, the new President nominates someone for the Vice Presidency who is then confirmed or rejected by the Senate, and if confirmed, then appointed by the President?

Edit: To clarify, I mean pre-25th amendment.