Does it really make sense to decide an otherwise drawn game with relatively even clocks based purely on who can push wood faster?
Does it really make sense to encourage or incentivise players to make the attempt at the expense of consideration for their opponent's time?
Deciding tournament wins with "armageddon games" seems to be rewarding precisely the player who "pushes wood" faster. We can decide that we don't want to do that, but then I think we need to change the nature of the game being played, which has been suggested re adding time per move. If you are playing a lightning fast, sudden death game, then the fastest wood pusher may well be the ultimate winner.
I suppose the logic would be that it's respectful of your opponent's time to allow the draw. If I have a 5 min to 2 min time advantage and try to blitz out the win, I probably have a less than 1% chance of winning in that game. I even have some chance of losing I suppose. If I have a 1 min to 10 sec time advantage though, I have a much better chance of winning. So I don't know if these cutoffs are arbitrary, but they may be personal.
The key item is that sportsmanship is often related to respecting your opponent's time in chess. If I'm in a turn-based game which is an obvious draw, I might look at my opponent's available vacation time and then try to figure out the chance that he ends up losing on time because he doesn't have any vacation time left. Maybe if he has 30 days left, like me, I take a draw. Maybe if he has 23 hrs left, I might think it's a good idea to prolong the game as long as possible, hoping he'll have a real life event which forces him to lose many games on time. Or maybe he'll just grow bored with having to login every 3 days just to make moves here. Maybe he gets burned out. One can make whatever argument one wants, but the relationship of respect for opponent's time and sportsmanship is very clear.