Wasn't ARPANET started as a project where individual communication nodes could fail, and the message could still arrive? Military technology generally needs to be reliable, which makes it a good fit for civilian use as well. Of course, there are plenty of aspects of military technology that isn't as necessary in civilian life.
It was originally an internal tool for ARPA researchers to collaborate. It had very little to do with the military, and ARPA scientists were not the only ones working on something like this. It absolutely would have happened somewhere else around the same time if the defense department wasn't interested.
