Why did Dennis Lynds and Kin Platt use pseudonyms when writing books for the Three Investigators series?
Using pseudonyms is a common practice among authors for a variety of reasons. Some authors want to preserve their anonymity; others want to write in several different areas and use a different name for each genre, and there are probably many other reasons.
There is a little speculation involved here, but not a whole lot. Dennis Lynds was a prolific writer who authored books under several names. He explains in an interview that he wanted to use a different name for the T3I books to distinguish from the pseudonyms that he used for his adult mystery novels. You can read the interview about his various names and the way he came up with the name "William Arden" at my friend Seth Smolinske's web site here.
Kin Platt's story is a little harder to reconstruct, because he was a well-known children's mystery storywriter already. So there does not seem to have been as much reason to use a pseudonym on the surface to distinguish from writing in other areas. However, on closer inspection, it's probably not too farfetched that Platt may have felt that writing books for a series (especially for a series that he did not create) was not as acceptable (from a literary perspective) as writing his own original work. Platt had just won the Edgar Award (akin to the Oscars for mystery writers) for Best Juvenile Story in 1967 for his book "Sinbad and Me" and had been nominated for another Edgar in 1970; so he may have been concerned about his reputation. The origin of his pseudonym for T3I stories (Nick West) is unknown, but we can make some guesses. "Nick," of course, is pronounced the same as "Kin" spelled backwards; and "West" may have been a nod to two of Robert Arthur's pseudonyms (John West and Andrew West).
Of course, Robert Arthur used pseudonyms in some of his non-T3I work. You can read about those at the offical Three Investigators site by clicking here.