The short version is the software might be too old on the 2013 Fire tablets and Amazon doesn’t seem to be updating them anymore. I’d like to dig into it a bit more to see what it might be missing and if I can work around it though. 
Long Version
I did a bit more research and it looks like the newer 7" models are just part of the “Kindle Fire” line whereas the 3rd and 4th generation like you have were technically part of the “Fire HD” line which is only available in 8" and 10" models in the most recent years.
Either way, the key thing that matters is the software version (which you very helpfully provided in your original post)! It looks like Fire OS 4.5.5.3 is equivalent to Android 4.4 (KitKat) and unfortunately that’s the latest version of software provided for the 2013 devices.
Silk is Amazon’s custom browser, so they may be able to keep it up to date via the Amazon App Store which would explain why it works. Unfortunately, Amazon has a tendency of not keeping things like the built-in “Webview” up to date. And since the webview is used for rendering webpages in browsers like Fully Kiosk Browser and other third party browsers, that explains why it might not work on an older Fire OS version.
We included some changes in the last two SharpTools.io platform releases to improve support back to Android 4.4 - now that I think about it @Steve_Jackson also mentioned that controlling things still wasn’t working for him on his older Motorola Xoom with Android 4.4.4 so it may be something that I can reproduce via an emulator and then see if I can work around it. 