Yeah I'm not sure why they think the Navy is doing it… I think the Navy is "just" the customer. Of course there's a lot involved in being the customer too.
I don't think the Navy is "just" the customer in a normal DOD contracting sense. It looks like more joint research and DOE involvement. Plus, mix it multiple contractors and locations and you have a very complicated chain.
I think the premise is that the US Navy is quite good at building nuclear power plants as multiple reactors power each and every submarine and Aircraft carrier we have. By the numbers the Navy has more nuclear powered ships than the US has nuclear power plants. There's also precedent for smaller scale plants that are lower maintenance and less dangerous.
I wouldn't necessarily say they are less dangerous. The designs can be very different and might even require more maintenance and safely procedures than the land based ones.
Also, the Navy isn't the the one actually building it. The contracting companies do that. Many of the people who run the systems in the Navy leave and become contractors/consultants for the higher pay.