Fascinating! The story of the Admiral's Rose Bush highlights a very distinct difference in British and American culture and perhaps even the real roots of the American Revolution and the drive to move westward from the eastern seaboard. A significant majority of folks over here in the colonies have little truck with aristocracy. Military rank is a job, and only to much weaker degree is it a social position.
The US military has no-fraternization rules as well, of course, but they apply more to romantic situations than being courteous and neighborly and even then the rules generally come into play only if it happens in the chain of command or involves harrassment. I'm sure some US admirals would have pointed the direction and told you to take a hike, but most of the flag officers I've met personally would think it abhorently discourteous not to offer a ride to an able seaman in that situation. The thought that he might be undermining discipline in Her Majesty's Navy probably never crossed his mind.
I'm tempted to speculate that the US take on these things is more effective, based on the track record of the US water company. There have been many battles where an American fleet commander took advantage of a foe's rigid chain of command to win the day even in the face of unfavorable odds (including squabbles with the British navy but notably against the Japanese in WWII). The same goes for ground and air operations. But since WWII, I suspect that the success of the American military owes a great deal to the quality of weapons and training.
On the third hand, today's weapons are increasingly making each individual soldier more and more an independent operator who has to be trained to think and act independently. Even more so for aircraft pilots. Each soldier has to be able to replan the battle and invent new tactics in real time, which drives toward a more level social stata. Flag officers pass along policy and plan strategy, but the less they meddle at the tactical level, the more successful the operation will be.
Wild thought: Imagine a platoon of fully equipped US infantry against all of the Roman legions that existed at the time of Julius Caesar. The legions would line up to march, the Americans would take out their officers, and with no one to tell the legions to keep marching, that would be the end of it!