The solution is not to install macOS on an external drive but to move your data to an external drive. Things that suck up tons of space on your drive like music, movies, TV shows, photos can easily be moved to an external drive and there is no reason for them to be required to be on your boot drive:
Change where your music files are stored on Mac - Apple Support
You can simply move your entire Photos library to an external drive and hold down the shift key the next time you open Photos and select the new location.
You can also move any of other data to the external drive as well.
Once satisfied that all the stuff on the external is accessible, delete the items on your internal drive and viola! more space.