You can add tunes that are not in Apple Music. You can add them as lossless. Yes they upload as lossy and will play as losssy when playing from an iOS device. But so much will still play as lossless, if not hi-res lossless, if Apple has those versions.
Myself, I mostly play from my iMac over home studio so everything is ALAC.
But when playing from my iPhone through Airpods or connected to car stereo there is no way my ears are not golden enough to discern lossy v lossless. Are yours? Truly? Even if you're young? Or do you listen through super high quality over-the-ear headphones or in-ear monitors?
I wonder if something like iMazing could get files onto your phone. Normally it is used to get media off a device.
I know you'd rather not but I'm thinking your only real solution is to use a 3rd party app. How large is your your device's storage?