Now that I think upon this a bit more, I believe the problem isn't so much speed, nor even team power ratings.
The problem is how huge of a difference minor changes in speed make. Literally having a single extra speed over your opponent could be the difference between winning and losing. No other stat suffers from this problem.
I don't have any great solution for this at the moment, but it almost feels like there should be some variability in how speed works. Literally every other stat has ranges - the same attack from the same champion against the same champion with the same stats and buffs will result in different amounts of damage being done, because of those ranges.
I wonder if speed worked similarly, if we'd see this be less of a problem - simply because teams wouldn't be able to rely exclusively on speed to win fights anymore. I know this would completely mess up CB and other places, so perhaps if they just changed arena specifically, to make it so that every turn, your TM fill rate is based on your speed multiplied by a random weight - between, say, 0.9 and 1.1 - maybe that would force teams to not just focus on getting the perfect speed balance?