How do people manage recommendation letters?
No it is not rude but there is a matter of fatigue in the end. the professor might just say "enough" at some point. Usually the letters will also be fairly similar so that each is largely composed of the same text but with smaller alterations tailored to the specific job application. Another trick is to ask the applicant to write a basic text covering what the applicant wishes the letter to cover. It is then easier to use that text and make modifications as one pleases.
Hopefully the applicant has more than one person to ask for references. Writing such letters is, at least in my sphere, something you are supposed to do, unless it inflicts severe effects on other activities. Yes, it is annoying at times,particularly if the request is such that one needs to complete the letter "yesterday". Hence providing some time will alleviate the discomfort.