In reply to Blue Straggler:
Google's website version of the Play Store that you're visiting on the Kindle Fire isn't designed for downloading and installing apps directly, it's for remote installation onto other devices that you own. Logging in using your own Google account will, by design, only display devices that are associated with your Google Account (i.e. Android devices with the Play Store app installed on them and your Google account logged in on the device).
The simplest option here would probably be to install the Nest app from a direct download from APK mirror: https://www.apkmirror.com/apk/nest-labs-inc/nest/nest-5-30-0-10-release/
The downside of that option is that the app won't update automatically so you would have to go back to APK mirror periodically to check for new releases. If you didn't, it would probably stop working eventually. That might be an ongoing frustration for you as designated family tech support!
Another option would be to install the Google Play Store. There are some instructions here: https://www.howtogeek.com/232726/how-to-install-the-google-play-store-on-yo...
It's a much more involved process but it might make other apps easier to find in the future as well.
It's frustrating but it's the business model that lets Amazon sell their tablets so cheap. They do their best to keep you largely confined to their own ecosystem of apps so that sales through those effectively subsidise the tablet.