The original comment was about the "US military" and not just the Navy. I'm assuming you're speaking from personal experience and that's why you mention the Navy specifically. The Navy does a great job, especially as it pertains to complex industrial systems. However, each branch and each unit has a unique culture and from my personal experience, the closer one gets to combat arms, the less bureaucratic and more "flexible" people tend to get with procedure.
Maybe your criteria is different for what constitutes a mishap, but the Navy has had what I think from a safety engineering standpoint would count as mishaps or at the very least close-calls. From just one decade:
December 12, 1971 – New London, Connecticut, US – Spill of irradiated water
During the transfer of radioactive coolant water from the submarine USS Dace to the submarine tender USS Fulton 500 US gallons (1,900 l; 420 imp gal)
were spilled into the Thames River (USA).
1975 – location unknown – Contamination
Radioactive resin contaminates the American Sturgeon-class submarine USS Guardfish after wind unexpectedly
blows the powder back towards the ship. The resin is used to remove dissolved radioactive minerals and particles
from the primary coolant loops of submarines.
October 1975 – Apra Harbor, Guam – Spill of irradiated water
While disabled, the submarine tender USS Proteus discharged radioactive coolant water. A Geiger counter at two of the harbor's
public beaches showed 100 millirems/hour, fifty times the allowable dose.
May 22, 1978 – near Puget Sound, Washington, United States – Spill of irradiated water
A valve was mistakenly opened aboard the submarine USS Puffer releasing up to 500 US gallons (1,900 l; 420 imp gal) of radioactive water
Maybe you were only considering nuclear reactor incidents? I can understand that perspective even if it probably doesn't meet safety engineering principles.
The numbers are greater if we use the OP's scope of "US military" and greater still if we count National Laboratories doing military funded work. I don't say this to disparage the military's competence, I actually think they are very good at managing complex systems. My main point is that I'm not convinced it's necessarily a byproduct of organizational structure of "chain of command" as the OP said.