I don't agree with that language. What if a runner is more than half way to a base where the OBS was just tagged out? You are going to award a runner not affected by the OBS a base to which they would most likely be put out?

This was the interpretation offered long before 95.

Okay.

Thanks all for the clarification. The final answer to my OP is, "because it's the rule."

It seems to me, the point of killing it at the time the OBS runner is put out, is to let the dust settle and let the viewers know the result, while still potentially depriving the offense because of an infraction on the defense's part.

All the other DDB plays let the offense get as much as they can, and wait until play has stopped, before ruling.

Is the OBS rule the way it is because it's unfair to make the DEF mistakenly think they have an out (maybe a 3rd out) while allowing runners to keep going?

I always figured it was because umpires could barely handle the rule as is. And killing it makes it simpler to unwind.
Consider: Bases loaded, no outs. B4 hits an easy double play ball to short. While F6 misplays the ball, F3 obstructs R3 going toward second. In avoiding interfering with F6, R2 trips on the way to third. Due to the delay caused by the obstruction F4 is still able to tag out R3 after F6 recovers.

Normally, the play ends here and we make a fairly straightforward award.
But now you are about to enter another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land of imagination. Next stop, the Twilight Zone!

Obstruction is a ddb here.

F4 steps up and throws to F5 who is standing on third base. With R3 called out F5 starts to chase R2 back toward second. A rundown ensues. B4 scampers up toward second in the confusion. R3 is tagged out going toward second and the ball is overthrown into right field. B4 advances to third. The ball is returned to the pitcher and you call the ball dead. Award bases as appropriate. And don't even get me started on if something crazy had happened.

Sometimes INT deprives the defense of a double play (sitch: R1 is hit by a batted ball to F4 while running to 2B). In this case, we get an out and the BR gets 1B. "That's just the rule."

How can R3 be tagged out on a ball thrown into RF? Especially when he was already called out somehow by F5 standing on 3B.

OBS is the 3rd easiest rule to understand, right behind IF and base awards

R3 wasn't out when F5 stood on 3B because R2 was out at 2nd so it wasn't a force. The throw to second came after the tag out trying to get the BR who was advancing on the rundown.

It's only an easy rule to understand because we don't let weird stuff happen after the ball is declared dead. In the twilight zone it's a very complicated mess of a rule.