I've been using Calibre to reformat books to MOBI format, then sending them to my Kindle library using Send to Kindle. From my Paperwhite I can see them in my library and read them all fine. But in Kindle for PC, those newly added books are missing from its view of my library.
Is there some kind of permissions barrier that prevents those books from being visible in Kindle for PC?
yes, Kindle for PC only officially supports books purchased from Amazon. Ditto for Kindle for Mac and the Cloud Reader. you can import pdfs to Kindle for PC/Mac or else copy your mobi files to the storage location for your downloaded books and access them through Kindle for PC/Mac but they will not sync with your account/other devices and your notes an highlghts will not be stored on your account.
I open non-Amazon mobi files (from a variety of sources) in K4PC all the time, mostly as a check that the files are good/readable. That auto-copies them to the K4PC location; however, after I confirm the file is OK I often just delete it from K4PC -- not from my own file-stash! Sometimes (increasingly often) I may then send it to my Cloud library.
I don't use Calibre and the only converting I do is the very occasional Convert when sending a PDF to Amazon Cloud.
I do almost all my PDF reading on either a PC or an iOS device PDF app that is made-for-purpose.