With the release of WatchOS 4.3 last week, the ability to browse your iPhone's music library from your wrist has been restored.
Before you can begin using the (sort of) new feature, you'll need to update your Apple Watch to WatchOS 4.3. To do that, connect your watch to its charger, and open the Watch app on your iPhone. Go to General > Software Update then follow the prompts.
Once the update is installed, open the Music app and scroll to the top, going past all of the album art. Eventually, you'll find a button titled iPhone. Tap on it to view your music library and, using your Watch as a remote control, begin playing music through your iPhone.