Yes, but...
The question of whether the Internet is essential infrastructure that should be run as a public utility does not resolve the question of net neutrality. It simply changes the process by which the question gets resolved.
In fact, if the internet was run as a public utility, I think that it would be less likely to support net neutrality, for 2 reasons. First, net neutrality tends to level the playing field between large companies and small start ups. However, large companies tend to have much more political power than smaller companies, so if the question of net neutrality was determined purely in the political realm then net neutrality opponents would appear to have an advantage.
Second, net neutrality tends to favor content owners over distribution channels. If content owners were still private companies, but the distribution channel was publicly owned, I think the public would tend to side more with giving power to the publicly owned internet utility companies and demand that companies like Disney or Google or What's App pay to play.
However, in a world where the benefits of getting rid of net neutrality went to your local city instead of, say, Comcast, the decisionmaking calculus of whether net neutrality is a good idea or not might change substantially.