How can I listen to my purchased music without iTunes on iPhone?

1. iTunes Match is still an option. You can re-subscribe if you need to. Don't take too long as without an active subscription music in the iCloud music library will be removed.
2. Music on the iPhone can be synced from a computer if you don't wish to use iTunes Match or Apple Music.
3. If your music came from either subscription service but was not actually downloaded to the device, then there's no way to access it on the iPhone without either the subscription to either service or syncing from a computer.
4. If the music exists on a computer then you can sync it form there onto the iPhone and select specific playlists or content to sync and you won't require a subscription to either service.
   1. click here ➜ Sync music between your Windows device and iPhone, iPad, or iPod - Apple Support
   2. or here ➜ Sync music between your Mac and iPhone, iPad, or iPod touch - Apple Support

If you no longer have an active subscription to iTunes Match, you can't access the music there. It requires the subscription for that.

You need to sync it from a computer in that case

You also need to make sure Sync Library is turned off on both the iPhone and the computer that still holds the music.

On the iPhone go to Settings ➜ Apps ➜ Music ➜ Sync Library and turn it off if its still on.

If the Music is not available on the computer then you cannot sync it to the iPhone.

You should not need to restore the iPhone, but if it still won't sync, then you may need to.