Winning the round before all players have had a turn in Phase 10
There is nothing in the rules about this. Yes, a round could end before a player gets a chance to play.

This isn't really any less fair than anytime a player other than the last in turn order ends up going out; in that case the last player has less turns than anyone else. Having 0 turns instead of 1 isn't all that different from having 4 turns instead of 5. So if you wanted a house rule to make it more fair, you should always end with the player who played last; not only if it happens in the first round. 

The extra advantage of being the start player is balanced out by having the start player rotate.