If you think the Klingons, even the smooth-forehead variety seen in TOS, would allow someone else to do the fighting for them against their enemies, then you are not talking about the same Klingons that have been shown on screen over the decades.

Except for when they would allow other people to do the fighting for them.

Part of the premise of A Private Little War was that there was fears that the indigenous people of Neural were going to come under the influence of the Klingon Empire. Surely this was meant as an invitation for the Federation to get involved in a proxy war, and if I recall correctly, Kirk even gave some of the tribes equipment so they could start to fight off the Klingon-backed tribes.

Tell me, what happens in a proxy war? One nation or coalition of nations back one side of a war, and another nation or coalition back the other. That's what happened on Neural.

That's not something that can be learned through proxy wars or cold wars. It's something you only see in battle against the enemy.

Based on the canon material available, there were really only a few times when relations between the Klingons and the Federation collapsed so far that they became a shooting war. Errand of Mercy was one of the times when things devolved this far; and I think that during The Search for Spock and The Voyage Home, things got pretty close to becoming a shooting war.

Really the only time I can think of when relations between the Klingons and the Federation are referred to as a state of constant warring is in The Undiscovered Country. But this was in a Starfleet meeting; this could just as easily be because they understood when and where the Federation and the Klingons were backing different sides of a conflict. It could also mean that the cold war between the Klingons and the Federation occasionally turned hot; or it could be a mix of the two.