You're wrong about
why the US Navy is doing that. They don't care about how those missiles were manufactured, they want to figure out what their performance is – what's their real fuel capacity, their real fuel efficiency, their real turn rates, etc. pp., to figure out how dangerous they are, and how they're best intercepted. Whatever the official performance figures are, they're guaranteed to be wrong (either exaggerated to scare enemies, or downplayed to lure enemies into a false sense of security).
None of that applies to civilian rockets. ITAR regulates them because exporting either the finished rockets, or their components, or the tools and processes used to make them, might help foreign powers build their own. Salvaging wrecks isn't very useful for that. Meanwhile, if you want to know the exact performance figures for those rockets, you can just look them up on the manufacturers' websites. Nobody has any reason to lie about them; if you exaggerate them, customers will find out quickly, and if you downplay them, customers won't buy it.
This is why the Chinese and Russians (and before them, Soviets) tend to keep "fishing trawlers" full of SIGINT and recovery gear around NATO missile exercises, but not civilian launches. They care just as much about NATO military missile performance as the USN cares about their own. (And both sides rely on good old industrial espionage to copy manufacturing technology.)