It tells people currently on the crossing how long they have. If they're infirm, they may be better off waiting at a traffic island than trying to rush across. The red hand is pretty clear - it means "don't start crossing", not "start crossing if you're confident you can make it in time" (though that is the way the countdown is used).
$200 is a ridiculous fine, though - that's over 20 hours work at the minimum wage.
Though I agree - there should be some way to indicate "do not attempt crossing, even if you think you're so fast." Perhaps adjust the countdown and keep the hand for several seconds more than it is currently.
The countdown allows me to choose to cross if there are 20 sec left (enough time for me to get across anything less than an 8-lane highway) or not to cross till the next light if there are only 3 sec left when I reach the curb.
With old-school flashing lights all I know is that the light will change soon, but gives me no idea HOW soon. In pedestrian friendly cities generally people will cross if it's flashing (esp. on a narrower street) because they have right of way. Heck, most of us will cross a one-way street AGAINST the light if the nearest oncoming car is a third of the way down the block. That's not the least bit dangerous and doesn't interrupt the flow of traffic.
I agree that LA shouldn't have bothered with the countdowns if they're going to consider starting across the intersection after the countdown has begun to be jaywalking.
LA was the only place I've ever been crossing the road with the valid pedestrian signal and had to stop walking or I'd literally walk into the car turning across my path (to the right, with the same green light I'm crossing with). It happened a couple of times, and I was only there for one week.
You also clearly haven't spent much time in L.A...many of these signs flash the walk signal for literally 2 or 3 seconds before beginning the 20 second red countdown.