No - this is false.
It was the existential and immense nature of the consequences of war that enabled the parties to 'back down'.
Without nukes - the US and USSR would have likely started a bunch of scuffles here and there.
There's a great chance that US/USSR navies duked it out over Cuba - because the consequences would have been likely very limited - unless they escalated of course.
Yes, we were lucky that it didn't go down as well, but nukes have been a huge factor in keeping the war basically cold..
"some less risky peace-building efforts" - there is really no such thing. 'Balancing Power' is what creates peace for the most part.