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You can sideload most any Android app, and as long as you have a mouse, you can use them, albeit they are often rather clunky. Only Android TV apps are available from the Play Store, however. I certainly wouldn't try to install stock Android vs Android TV if that is what you are after.
You can root the Shield, and there are third-party ROMS available for it. You can sideload standard Android apps without having to do either of that, though; you just need Sideload Launcher or the equivalent to launch those apps that don't show up in Leanback, and you even get mouse emulation using the right stick of the game controller.
To be honest, though, the Android TV app situation is nowhere near as dire as it once was, and generally, sticking to Android TV apps is going to get you a better (more living-room focused) experience. The Shield even has an official Amazon Video app, which was a major coup for them. It's pretty much only them and Sony who have pulled that off at this point.
A better question might be what apps you are looking for that aren't on android TV?
It's not the Nvidia version, it's Android TV. What you had before is a box that runs the same Android found on phones and tablets which is why you could access all the apps on the store. Android TV limits it to those with the TV UI (Leanback), but you can easily sideload apps to it, although they won't be optimised for the remote.