Penalty kicks should not determine the outcome of a game. I played in high school, a long time ago. I'd remove a player from each side every 5 minutes in overtime and play until a goal is scored. Only in tournament games, of course. A tie is a perfectly reasonable result in league games.
The NHL needs to eliminate the shootout too. A tie is fine after one short OT period, 4 on 4, but not necessary for league games.
In playoff games, they should play one 10 minute OT, then remove a player from each side, play 10 more minutes, then play 3 on 3 until somebody scores. Somebody WILL score.
Any other problems that need solving?