We're arguing about the high-level outcomes without looking at lower-level what is actually happening here.
Historically I think it would be hard to argue that the NPT really stops countries from developing nuclear weapons if they want to. They either build them outside of the agreement, or violate the agreement. Similarly other arms control agreements don't happen in a vacuum.
That said, the NPT exists in a broad international political situation in which there is widespread acceptance that proliferation is undesirable, in which most of the countries with ample capacity to quickly develop such weapons didn't need to because of a security umbrella provided by the U.S. or previously the USSR, etc. If you still have all of those tenets in place then possibly the situation wouldn't be all that different just because the actual NPT talks broke down or didn't happen.
On the other hand, if they don't happen because the international situation is sufficiently volatile that the U.S. and the USSR can't persuade other countries either that they're safe under a security umbrella or at least that as neutrals the nuclear weapons would never be directed towards them anyways, then the Cold War international order has already broken down on some much deeper level and yes, I think you would expect in that scenario to see far more nuclear proliferation.
I don't want to draw us into contemporary political debates overmuch, but it's worth bearing in mind that both today and in the past, those countries who chose to buck the anti-proliferation consensus were on the fringes of world politics and therefore, at least in their own minds, in the greatest need of some sort of ultimate guarantor of security because they couldn't count on others to protect them -- e.g. North Korea, Iran, Iraq, apartheid South Africa, and obviously in a different camp entirely but for weirdly similar reasons, Israel. The more unstable the world is, the more countries will fall into that basket of feeling isolated and seeking protection.
Relating back to U.S.-Soviet relations, if that relationship is volatile enough and unstable enough that you're having all those other knock-on effects in terms of nuclear proliferation, then it would be hard to imagine them trusting each other enough to undertake serious arms control talks, either.
