It's hard to be specific without a specific situation to address. Combine that with I don't have specific experience with the Rewrite API myself, though I do understand the concept well enough. I'll do the best i can.
Using your hypothetical example, your goal is to use the API so that an entry ends up in the .htaccess file that modifies any url with "/people" as the first folder argument to be made into "/about-us/people" behind the scenes (meaning the url in the address bar does not change) with anything that comes afterwards to remain as-is.
You could maybe do this with an .htaccess entry alone. The reason for using the API is the query variables get handled properly, but in this case, I believe this will happen in either case.
That leaves the navigation issues, which is partly dependent on the specific templates involved, but shouldn't be that difficult to alter. The other navigation issue is WP functions that output links. In this case, there is typically a filter you can hook to take out the unnecessary "/about-us" url element, or there is a variant function that returns a link instead of echoing it, giving you the opportunity once again to remove the "/about-us" element.