Using this online compiler, the only strict problem I had was the Kindle's refusal to open PDFs online, as you found. It would be fairly simple for someone running a similar site to automate sending an email with the PDF as attachment rather than linking to the attachment direct, and you can set up your Amazon account to deliver PDF attachments to the Kindle. I don't know much about the way Kindles are set up, but if it were possible to bypass that restriction (which, given that Kindles can handle PDFs, and can download other types of files, feels like it should be possible) that would fit the bill.
Having said that, even the almost negligible amount of code I wrote on that site on my Kindle was incredibly laborious; I used my Kindle for internet for several weeks when my laptop broke a few years ago, but the quantity of backslashes and curly brackets needed for LaTeX make Kindles singularly unsuited to the job, in my opinion. Even if you were only editing documents the amount of time it takes to scroll up and down would be maddening.