Apple TV will work and will do what you want assuming the home theatre receiver has HDMI or Optical Digital Inputs. It will work and Apple TVs are very cheap but feels less than optimal as this would not be through the same system as your vinyl.

You could run an optical cable out of the Apple TV to an external DAC that attaches to your integrated amp or a DAC in your amp and you would use your iPhone or iTunes on your computer as the controller via Airplay. This would have your audio going through the same system as your Vinyl.

IMO, the best solution is to get an AirPlay enabled network player or receiver which will allow you to stream files from your PC or phone, has an exceptional DAC and usually a few other functional services that make it a really nice service. I have this functionality in all 4 of the systems in my house. However this is the most expensive solution.

For instance, you could get a Marantz M-CR510 which is an inexpensive 2 channel network receiver that is AirPlay enabled and streams Pandora and Spotify. It has one analog input so you can plug your phono stage from your turntable into it. This is $499 and I have one in my bedroom and it sounds great.

Alternatively, you could get an integrated amp like a Music Hall 15.3a or a Rega Brio R and an external network player like a Pioneer N-50 for $699 or a Marantz NA8005 for $1199 are amazing units that are AirPlay enabled that would serve as an external DAC and streamer. I have a Rega Brio R and a Marantz NA7004 which is now discontinued as the guts of my primary 2 channel audio system.

My other two systems are multi-channel home theatres with an AirPlay enabled receiver or pre/pro.

The upside of the Apple TV is that it has the optical and HDMI outputs. Enjoy and good luck

i'm curious do you need to have a monitor (TV) hooked up to the apple TV when connected to your system or can you see everything you need on your iPhone controlling it?

if you are just going to Stream Audio via AirPlay, then you do not need a video monitor. To use any other feature, you need a video monitor.

Hey Dave!
I tried the Apple TV option for a while and it worked pretty well for files stored locally but struggled a lot with services like Spotify and anything that required an internet connection.

I guess it depends on your Internet speed but I have now installed a Bluetooth streamer and works great! Mine's Peachtree BT1 but I'm sure there are other options, this one offers a Digital output, as well as an Analogue one and a DAC. It costs pretty much the same as an Apple Tv. You can easily use it with your iPhone.

Hope it helps!