Interesting options here suggest I'm not a professional but in my opinion all the options are good)
On the contrary, all dedicated options are quite poor. For a gaming console, the Android OS just isn't there, yet.
The controllers that transform your phone into a mobile gaming display are nifty, but while the Tegra K1 & X1 processors in the NVIDIA Shield both have ample horsepower to motor serious games, certainly decent even if they aren't up to Console/PC flagship quality, there's still barely any games made for Android that utilize controllers, not simple touchscreen controls, and almost all of the games on this operating system are either free and ad-laden, or operating on a freemium model. Very few good games charge you up front and deliver a worthwhile product.
Considering that you get far, far more processing power, an older, more stable, and more secure OS, far better games, more of them, superior online gaming communities, and all sorts of online goodies and services that aren't available on Android, it makes absolutely
zero sense to purchase an NVIDIA Shield 16GB for $200 or 500GB for $300 when there is an Xbox One or PS4 (1TB of either) bundled with one of the most popular AAA games at any given moment for ~$250. Zero sense. It's a terrible purchase.
Similarly, the Apple TV (4th Gen) might be a great product, but not for games. The special attention that Apple software gets means that their superb remote controller gets extra focus for many games, but that's still a far cry from a dedicated dual-analog controller. The A8 processor is respectable, but even weaker than the Tegra processors, and the Apple TV only comes in 16GB or 32GB variants making serious game storage implausible.
I still strongly favor the baseline Xbox One and PS4 as the market's entry HTPC offerings. Just look past the Fire TV and the rest of the dregs.