Streaming Music through my home stereo

The simple solution would be to stream wirelessly via Bluetooth from the iPad to a Bluetooth receiver plugged into the Pharaoh using a stereo mini to dual RCA cable. That limits the quality of any files that are of higher resolution than MP3 or Apple Music's proprietary AAC format but if Apple Music is your primary source the quality should be quite acceptable.

Here are a couple of well-reviewed options at different prices. The Audioengine B1 has the advantage of using Bluetooth with APTX which should make a small but noticeable difference.

https://www.logitech.com/en-us/product/bluetooth-audio-adapter

https://audioengineusa.com/shop/adapters/b1-bluetooth-music-receiver/

BluOS would certainly be a good choice for most music services but since Apple doesn't yet provide an API that would allow access by Blu it isn't an answer for someone who wants to use Apple Music. There are definitely other excellent choices of music services.

Yes, Apple Music is compressed, just like Spotify and Pandora. The advantage of any of those services is lower cost and sometimes greater convenience compared to higher fidelity options, considerations of the different music catalogs aside.