The temporal cold war is just a conflict that took place somewhen. It has nothing to do with the existence of the agency that is protecting the original timeline, nowhere it says that it would stop existing after the cold war. Especially after the war having control over the timeline would make sense to prevent another war. Apart from the fact that "before" and "after" hardly have any meaning if you have access to time travel.
And if they agree that the future would be so bad without a change of the timeline, why not do it themselves in a controlled environment than relying on some random eperiment that already cost many lives (and therfor already has drastically changed the timeline)? It's just another thing they did not think through, or cared about. Not necessarily now, but when they introduced such an institution in voy and ent. But they made it canon.