Ok here's my preliminary review after about two hours of play with my daughter. Keep in mind I have never tried Infinity on any other platform so I'm coming at this from an outside perspective. But I do own consoles (Xbox 360 and Xbox One) and have played a lot of video games over the years. I'm not really the type of person that would go for this type of game on my own (I'm more of a Call of Duty kind of guy). I bought it for my young daughter but will be playing it with her from time to time, much like I have played Minecraft with her.
Generally Infinity runs well on the ATV. I thought the graphics looked quite good and the framerate was fine (occasionally you'll run into a hitch but nothing major). The Nimbus controller is very good - the build quality is on par with an Xbox controller. Control responsiveness was immediate, just as I would expect. The Bluetooth Infinity base pairs seamlessly with the game (you don't need to pair it in the settings menu). Menus load quickly and everything works as it should. The overall experience is similar to playing on a console (with one caveat - see below). Though I have not tried Infinity on any other platform, it sure seemed like this was the full Infinity experience without anything stripped out or dumbed down for the ATV. There are lots of things to do and plenty of options in the menus.
Given that this is Apple's first shot at a TV box that plays games, I'd give it a thumbs-up.
However, Infinity puts the whole "content streaming approach" that the ATV uses, and your internet connection speed, to the test much more so than the typical ATV game like Crossy Road or Alto's Aventure. It's a much bigger game overall.
I'm on a 24 Mbps U-Verse connection, and loading time was a little (not a lot) annoying. There were occasional unexpected in-game pauses for streaming (I'm assuming these are one-time deals for each new area encountered). If your connection is less than 10 mbps, I don't think you should go this route, personally. Infinity is a big game.
Most content downloads occurred when loading a new level or area for the first time. The first time a new level is loaded it could take several minutes. After that, the next time you go to the same level or area, it's plenty quick.
In Toybox, I have encountered some sudden pauses for content downloads right in the middle of walking somewhere or when using a tool for the first time. I found that annoying. It's not well indicated on the screen and I thought the game froze at first. This seemed to become rare after playing for a while (presumably enough content was cached locally to cover everything).
Given the size and complexity of the overall game, I'm now starting to question choosing the 32 GB ATV model. This may be a case where 64 GB would have been a better choice. If I put two or three Infinity-size games on this box over the next few years, it'll be swapping content much more often. We'll see.
Overall I'd give this a thumbs up. Just understand the streaming aspect and make sure your internet connection is fast enough.