You’re correct that it wasn’t the initial prompting (convenience of using multiple interconnected terminals was):
“The ARPANET was not started to create a Command and Control System that would survive a nuclear attack, as many now claim. To build such a system was, clearly, a major military need, but it was not ARPA’s mission to do this; in fact, we would have been severely criticized had we tried. Rather, the ARPANET came out of our frustration that there were only a limited number of large, powerful research computers in the country, and that many research investigators, who should have access to them, were geographically separated from them.” ~ Charles Herzfeld
But the ability for the network to survive outages and attacks certainly was a goal as the network evolved in the context of military R&D. Nuclear strikes were a frequently cited example. It was developed during peak nuke-awareness — the Cuban Missile Crisis happened in 1962 and shocked the world so much that nuclear drills were common in schools decades later.
Arpanet was also partially funded from a ballistic missile defense budget:
His seminal moment came in 1966. He had just taken a new position at the Pentagon — director of the Information Processing Techniques Office, part of the Advanced Research Projects Agency, known as ARPA — and on his first day on the job it became immediately obvious to him what the office lacked and what it needed.
At the time, ARPA was funding three separate computer research projects and using three separate computer terminals to communicate with them. Mr. Taylor decided that the department needed a single computer network to connect each project with the others.
“I went to see Charlie Herzfeld, who was the head of ARPA, and laid the idea on him,” Mr. Taylor recalled in an interview with The Times. “He liked the idea immediately, and he took a million dollars out of the ballistic missile defense budget and put it into my budget right then and there.” He added, “The first funding came that month.”
According to Stephen J. Lukasik, who as Deputy Director and Director of DARPA (1967–1974) was “the person who signed most of the checks for Arpanet’s development”:
“The goal was to exploit new computer technologies to meet the needs of military command and control against nuclear threats, achieve survivable control of US nuclear forces, and improve military tactical and management decision making.”
“Why the Arpanet Was Built”. IEEE Annals of the History of Computing. 33 (3): 4–20
It’s easy to see how the “survive nukes” narrative evolved, and I think, obviously incorrect to say that surviving nukes was never a goal of Arpanet.