In this case, Wikipedia makes it pretty clear that while surviving a nuclear attack specifically might not have been an explicit part of the design goals of the designers (which may have started the myth myth), resilience against large-scale failures was exactly the feature that made the technology interesting to its military sponsors:
> Nonetheless, according to Stephen J. Lukasik, who as deputy director (1967–1970) and Director of DARPA (1970–1975) 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.
(https://en.wikipedia.org/wiki/ARPANET -- where the quote is sourced directly to an article by this Stephen Lukasik.)
My takeaway is that while the ARPANET creators might not themselves have particularly cared about surviving nuclear disasters, without the completely decentralized packet switching that made this at least a possibility, it wouldn't have ever received the funding to get off the ground.
Shortening this fact to "ARPANET was funded to survive a nuclear attack" is IMO a fair sound-bite sized summary, not a "myth" to be debunked, even though the story has obviously more nuances.