The current rules seem quite sensible to me.
Say I'm fighting my enemy Bob, and we have the same speed. Bob takes off running, and a split-second later, I take off after him. He started running first, so he gets a slight lead. Assuming we started out in adjacent squares, he is now a little more than 5' away from me-- just beyond reach of a melee weapon. So I can't hit him unless he slows down (or I somehow manage to speed up).
After a couple rounds of sprinting, we start making Con checks. If Bob fails first, he must slow down, and I can catch up and start hitting him. If I fail first, I can't keep up, and Bob probably gets away.
In a real-life chase, if Bob's Dexterity were higher than mine, maybe he could open up an early lead. But that wouldn't help much, since we're sprinting in a straight line, and I can keep coming after him. If he gets tired before I do, I can still catch up and start bashing him. In pursuit on open ground, stamina (represented by a Con check) is far more important than a short-term burst of speed.
[houserule]
As for diving tackles, I treat this as a special case of a grapple. I allow a grab attempt to be made during a sprint, as long as the target is denied his Dexterity bonus to AC. If the attacker fails, he falls prone, one square short of the target. If he succeeds, both characters fall prone in the target's square, and then they make opposed checks to see if the attacker can maintain his hold.
[/houserule]