I think the harm would be that it makes it more confusing, not less. Nothing could be simpler than furthest from the hole plays.
It would be a knee-jerk reaction to a "problem" that isn't really a problem. OP had a clueless playing competitor yesterday, and Annika goofed once a bazillion years ago.
You might be right for some and I agree that the current rule is very simple. While I typically advocate making rules simpler, my idea of simple is so that more people understand them intuitively. We have several rules in the game that make a distinction about being on and off the green- i.e. if you are on you can mark and clean your ball but not use the pin...if you are off you can use the pin but can't normally mark and clean your ball.
I have played with a lot of players who prefer to have everyone reach the green before pulling the flag and putting and have also played with some who prefer to have the guy farthest go first even if he is off. When you have a mix of philosophies, then yes, there is some confusion. If it is more efficient to have everyone reach the green before pulling the flag (as others stated above) I think this is the preferred method.
The current rule is simple, period. If a person can't understand it then he lacks the brainpower to find the golf course, much less play the game. It doesn't matter what " a lot of players... prefer". And it isn't more efficient if you know how to play. Here's the incredibly complex way I play it: I tend the flagstick for the person putting who is away, or if he prefers I remove and holding on to it I step a few feet away. He putts, I replace the flagstick, then the next player plays his shot from off the green. I remove the flagstick, or tend it if the next player so desires, and we all play out. No time has been lost, no inefficiency experienced, no play delayed. And we have followed what may be the simplest rule in the book to the letter.
You ask, ".... why does the order of play rule need to be different for match play than the way most people (who are concerned with speed) play?
The essence of match play is that your strategy is entirely dependent on your opponent's play. Unlike stroke play in which we compete against the golf course, in match play the match is, essentially, eighteen separate competitions. Par is meaningless in match play. In fact, no scorecard is necessary in match play.
Yes, but how does having the guy who is off go first eliminate match play strategy? Seems to me that while it might change who actually has to go first, both players would be subject to the same rules so if you want to putt last, you simply try to put it closer on the green than your opponent. Conversely, if you want to go first around the green, then miss the green or put in on the green farther from the hole than your opponent. That is a slight modification from how it currently works but I can't think of many instances where I would try to play a shot differently in match play because of this change.
How aggressive or safe one plays can be determined by how well his opponent does. Around the green, a lot of important information can be gained by going second, or one can put added pressure on one's opponent by making a good play ahead of him. How those factor into play depends on the situation. In match play I would never give up the order of play if it gave me a chance to gain an advantage.
I think certain rules could be simplified and/or made more intuitive and/or changed to help speed up play.
There is absolutely nothing unintuitive about "the player who is away plays first". As I showed in my comment above, pace of play isn't affected.