First thing, since you’re using iTunes to rip your CDs, I would highly recommend that you make sure you are using the "Apple Losseless Codec" when importing direct from CD. Otherwise, you’re losing a lot of resolution that you would never get back with upgraded equipment.
The others have good ideas for improvement: room treatment, interconnects, power cords, fuses. All those definitely will help.
In my opinion, your weak link is the AppleTV, as you have indicated. The Bluesound may be an improvement because you can use it’s toslink digital output to drive the Azure DAC.
Another way to go is to look at something like an Oppo BDP-103 (or the newer 203 when it comes out). It will likely be a much better transport for reading in the streaming audio data and converting that to straight PCM (including clock timing) and being able to use the coax digital output (which is generally better than toslink). The Oppo supports integration with Tidal. It will also play ALAC files if you want to keep using that. Then maybe look into JRiver software on your Mac as a DLNA server for the Oppo. Cost of the Oppo is about $499 or so (or less if used). You can also play straight from a CD or even USB stick/drive.