Maybe not quite the same thing - but have you checked out https://music.apple.com/
Open that in Edge, and then you can install that website as an App into your Start menu.
You can’t download tracks for offline play - but it’s definitely a lot ‘lighter’ than iTunes (and it’s in the same style as Apple Music on your phone, Apple TV, Macbook etc.).
(You can also do the same with Apple TV as well at https://tv.apple.com/)
Y̶e̶s̶,̶ ̶t̶h̶a̶t̶ ̶i̶s̶ ̶e̶x̶a̶c̶t̶l̶y̶ ̶w̶h̶a̶t̶ ̶I̶’̶m̶ ̶l̶o̶o̶k̶i̶n̶g̶ ̶f̶o̶r̶!̶ ̶T̶e̶s̶t̶i̶n̶g̶ ̶n̶o̶w̶ ̶a̶n̶d̶ ̶i̶t̶ ̶d̶o̶e̶s̶ ̶e̶x̶a̶c̶t̶l̶y̶ ̶t̶h̶e̶ ̶s̶a̶m̶e̶ ̶a̶s̶ ̶S̶p̶o̶t̶i̶f̶y̶ ̶C̶o̶n̶n̶e̶c̶t̶,̶ ̶b̶u̶t̶ ̶s̶p̶l̶i̶t̶ ̶i̶n̶t̶o̶ ̶a̶ ̶c̶o̶m̶p̶l̶e̶t̶e̶l̶y̶ ̶s̶t̶a̶n̶d̶a̶l̶o̶n̶e̶ ̶a̶p̶p̶.̶
EDIT: Okay I have to retract my statement above… it’s mostly what I was looking for, as I still can’t pick up a song on my PC (even with iTunes) that I started through the Apple Music phone app. You have to have iTunes desktop app already open so the iTunes Remote app can ‘see’ your media library and then you can go from there. So it really is an iTunes Remote, no integration with Apple Music at all!
Even if I have Home Sharing enabled on Apple Music (the same feature you have to enable in iTunes), it just can’t ‘see’ the library of the Apple Music phone app for some reason.
So still some way to go before it’s anywhere near the functionality of Spotify Connect.
Thanks for the heads up anyway!
I just wanted to say a massive THANK YOU - I’d completely forgotten this app existed and it’s exactly what I was looking for to control one from the other.
I’d really like to see this get to spotify level one day where I can start a song on Mac, pick up my phone and keep playing it on there…
But for now this is a really nice compromise
So I did a quick Google search for ‘Apple Music Windows’ today to see if there were any updates on release timeline. While I couldn’t see anything new about the final release, I did spot that the Windows Store download link for the official AM desktop preview has been reactivated.
Previously when I looked, the ‘Get in Store’ button was missing, as if they had paused the release.
Apple Music Preview - Microsoft Store Apps
Downloaded now and works perfectly for me. Most of the same features as iTunes such as Smart Playlists and local music, but the newer look of the iOS app. Also has support for Lossless for those of you with hardware that supports it!
One downside I just spotted: Because it’s not actually iTunes, the iTunes Remote app suggested above by jzw95 doesn’t work with it. Can’t have it all I guess… 
Also looks like the Apple Music App for Windows is only for Windows 11. If I go to the store to get it, on my Win 10 PC it just says I don’t meet the minimum requirements for it.
Can’t think of anything specifically in Windows 11 that’s missing in Windows 10 that would possibly break this though? (is there extra special encryption available in Win 11 that they might want the tracks to use to stop them being pirated?)