You're incredibly confused here.
They don't want to use Apple's infrastructure. Look at how Spotify Desktop works - you download it from their website, whether you're on OS X or Windows. They don't distribute Spotify Desktop through any app stores. Spotify Desktop is self-updating. It automatically checks their servers for updates and updates itself as necessary - the updates don't rely on anyone else's infrastructure. Payments are processed by them, not by Apple or anyone else if you subscribe through Spotify Desktop.
They use Apple's infrastructure not because they want to, but because Apple forces them to. They can't allow you to download the app through their own website, because Apple forces all apps to be installed via the App Store. They can't facilitate an autoupdate, because again, Apple forces all apps to be updated through the App Store. They can't process payments themselves, because Apple forces all IAPs to go through Apple's servers.
They have to pay to use Apple's services because Apple forces them to use Apple's services - they have zero interest in actually using their services, or else Spotify would choose to distribute the Desktop version of Spotify via the Mac App Store.
Apple isn't driving the price for the consumer down. They're driving the price of Spotify up by sticking their own fingers in the pie. Processing payments yourself costs around 2%. Hosting the app for download is completely negligible (especially when compared to streaming the music files, which puts strain on Spotify's servers, not Apple's.) So Apple is charging Spotify $3.90/month/customer for services that cost Apple around $0.26/month/customer. Which means for every Spotify customer who subscribes on iOS, Apple makes $3.64 in net profit, while Spotify's revenue (not net profit) is around $1.30/customer/month after paying all the labels and Apple. Spotify is literally making nothing because Apple is charging them outrageous fees for trivial services that Spotify would much rather do themselves.
Don't know about confused, but as you said Spotify does not want to use Apple's infrastructure.
That is their problem, so in order to reach Apple customers they have to make a choice, stay all windows and Android etc. or use the Apple structure to reach more customers.
Don't know whether all Windows and Android would generate them a profit.
Once they decided to use Apple, they have had to agree to Apple's terms, as frustrating as it is. It's Apples store, software base etc.etc., so again they dictate the terms.
Apple does not make 3.64
net profit, they do have costs and none of us know exactly how much.
Now, to whine that literally Spotify is making nothing, brings me back to : Why be in iOS or Apple structure?
It's as always a matter of money and somebody else trying to decide how much is fair or how much they should pay.
PS: I don't care about Apple music, as there are too many other options to listen to music.
PS#2: I sometimes download apps from websites unrelated to Apple, i.e. not going through the app store and they function just fine.
Just gives an" unidentified developer" warning upon installation. Would that not be an option? No idea if Apple is then getting a cut ?