just for airplay the aTV will work
the aTV 2 and 3 are basically the same. except for the aTV2 is 720 and the aTV3 is 1080
there is a TINY model number on the bottom, the 2 is A1378, the 3 is A1427/A1469
airplay tops out at 1080, so if you've got a 3, an aTV4 will not give you better quality airplay.
no one besides apple make an external airplay device, so if your tv doesn't have it built in, you must use the aTV.
all that said,
if you have another way to get the video to your TV, I would use it. Airplay is great in a pinch, but the quality can be lacking compared to other methods. especially if you're "mirroring" where the image appears on your device screen as well as the TV. If it can send the media directly (like some web pages) it's a bit better. But this all depends on what you're doing.
They've made the choice to prioritize stability over quality.
If you're playing locally stored media like TV or movies, check out Plex (there's an app for your TV) and a server program you run on your computer.
There are also a bunch of UPnP servers. Your TV should have a client already built in -
https://en.wikipedia.org/wiki/List_of_UPnP_AV_media_servers_and_clients
a "hard" cable will also give you better quality video,
they make lightning to HDMI adapters for your phone, or numerous options to connect to your computer.