Is there a way to send a music file from Finder to an iPhone's music library?
There isn’t a filesystem based local library on iOS. You will want to put it into Files app and use a different player if you really need to play files from an arbitrary box of files.

Files.app can play most songs without needing an app.

It multitasks and I have some FM radio recordings that are 2-3 hours long and they play amazingly well from Files app. I don‘t want these in my iTunes library so even though I’m all in with Apple Music and a cloud library - I have some esoteric stuff and audio books, lecture notes, radio performances I choose to store in the filesystem. Files on demand from iCloud works so well for my case I encourage anyone to try that.
Is there a reason why you’re using Music app and not syncing the libraries via Apple Music? Same question for using iCloud to sync files same as box.com / Dropbox / google and other file sync options between macOS and iOS.
It’s easy to automate moving files on macOS with services, folder actions and more, the problem here is Basically to chose a different player that works with files dumped in a directory as opposed to adding them to a music library that then syncs.
You are among some very smart and sophisticated People that have this understandable misconception about Photos / movies / books / music. Without an insight into how the ecosystem is designed, people revert to dumping files in a folder and hoping the apps read from a filesystem. No judgement, but this is a common request for something Apple design rejected about a decades ago as a sub-optimal experience on the platform. Since then Apple has doubled down time and time again on managed libraries and incremental sync. You don’t have to buy in to that, just find apps that buy in to how you like to store your data.