I think the OP's way of correcting things is fine. I think it would also be okay to be more direct about it. And I think it's just silly to pronounce something incorrectly, so others won't realize they've pronounced it incorrectly.
I would say, just make the correction natural. I mean, surely in the car on the way to the monument isn't the first time the monument's name has been said. Surely there's a point when you're discussing plans for the day, or the next day, and they say, "We'd like to go see <incorrect>." And then you could go, "Oh, <correct>. Yes, that's a great place to visit." And then every time you happen to say the name, you say it correctly; and if they can't or won't attempt to say it correctly in return, so be it.
If they're the sort to be offended by the suggestion that they don't do everything perfectly, I think there are bigger problems.
But, if they're getting everything wrong, maybe be judicious about which things you repeat correctly, and avoid repeating the rest. Museum they're going to be spending hours at? Worth correcting. Random street they're passing and won't pass again? Probably not worth correcting. My parents went on a trip to Hawaii and my mom had a terrible time with the local names. Someone who corrected every single thing, even in a polite way, would probably just have confused and stressed her. But, she would have appreciated hearing the correct names for major stops, even if she couldn't actually say them herself.