A peace treaty isn't a treaty of friendship or alliance. It just means there isn't a state of war between the groups anymore. The Organian treaty seems to have merely been a cease-fire - the war technically went on, with any contact assumed to be likely hostile. Even as the Ent-A was on a diplomatic mission, specifically meeting a specific ship at a specific time to specifically pick up the Klingon Chancellor specifically for a peaceful, diplomatic mission (specifically), there was a good bit of tension at first.
Knowing the Klingons, as soon as they got their own house in order, they would go back to their old ways. It seems the YE events were the normal order of things - eventually, the Klingons would deal with Praxis, build up their military again, and eventually clash with the Federation, because that's what Klingons do. I've never really thought about it before, but it's possible the Galaxy-class was designed so powerfully with full expectation of fighting the Klingons - that's why it's mostly the same in YE as it is in the normal show. But, the sacrifice of the Ent-C was so significant that it caused the Klingons to actually be impressed with Starfleet, and so the alliance was formed (as the Klingons will apparently be buddies with any honorable warriors, as long as they aren't actively fighting, which would explain the Klink-Rommie TOS alliance and subsequent falling apart); which seems to be pretty new in TNG, so it still took a while to get to that point. Without the anti-Klingon mission, the Galaxy could be repurposed into a more exploratory, diplomatic mission.
Also, they seem to think of treaties as mere suggestions, or relationships, not contracts. As soon as the Chancellor feels like the Empire has been slighted (dishonored), he can call off the treaty with a word - and then, once he's feeling better about the opponent, he will just state the treaty is back in force! (The diplomats and lawyers must really hate dealing with Klingons.) Even in TNG the Klingons in general were seen as a potential threat - not necessarily the whole Empire, but any BoP could have a captain that would do something stupid for glory and honor - and the Empire would disavow any responsibility.