So let's do away with a random way of deciding a winner that has little to do with soccer, and replace it with a random way of deciding a winner that has even less to do with soccer?
If something must be done, I think the best realistic proposal I've seen is incremental player reduction. Maybe after 90 minutes each side has to send a couple guys off so it becomes 9v9 (or less depending on red cards). At halftime in extra time another from each team for 8v8. Then if still tied, add on another 15 minutes at 7v7, etc.
The reason is that fewer players means more open space, means more goals. Sometimes you see this in games where each side has gotten red cards, it opens the game up a bit. Maybe also give each team an extra sub for each extra 15 minute interval, so fresh legs can exploit the extra open space.