The US also escalated at times, such as the reaction to missiles being placed in Cuba, which did almost provoke nuclear war.

And there was Nixon, who was busy trying to convince the world he was a madman willing to launch a nuclear strike to get what he wanted: https://en.wikipedia.org/wiki/Madman_theory#Richard_Nixon

Right, there was strife between the nations but also tension relief with some back channels remaining open and it was happening all at the same time.

Perhaps is just isn’t as visible now - but it seems like back channels keep getting intentionally shut.

What kind of face-saving option should we offer Putin? Keeping in mind that we don't run Ukraine's diplomacy for them...

The Cold War featured American and Soviet pilots shooting at each other during the Korean War, supplying arms isn't nearly as big an escalation as supplying fighters.

It’s not so much about face saving as it is about maintaining some form a transition path to a more peaceful state. And to understand what path might be an option one has to have some forum of frank discussion - likely not with the leaders themselves but with high enough level people from both sides that they are at high enough level to see the lay of land.