If we only handle AI that well doom is probable. It has economic uses, unlike nuclear weapons, so there will be a thriving black market dodging the safety concerns.
Also there are only a few companies that can fab the semiconductors needed for these training runs.
It is also quite possible for our society to decide that deep learning is too dangerous and to outlaw teaching and publishing about it, which would not completely stop the discovery of algorithmic deep-learning improvements (because some committed deep-learning enthusiasts would break the law) but would slow the discovery rate way, way down.
And most people who think AI "progress" is so dangerous that it must be stopped before it is too late have loose confidence intervals extending for at least a couple of decades (as opposed to just a few years) as to when it definitely becomes too late.
Unless the first one is so advanced no other can challenge it, that is.
Humans have prevented it many times, but not specifically by technological ability. If Putin/Trump/Xi Ping wanted a global nuclear war, they'd better have the means to launch the nukes themselves in secret because the chain of command will challenge them.
If an out-of-control AI could discover a circuitous way to access nukes, an antagonist AI of equal capabilities should be able to figure it out too, and warn the humans in the loop.
I agree that AI development should be made responsibly, but not all people do, and it's impossible to put the cat back in the bag. The limiting factor these days is hardware, as a true AGI will likely need even more of it than our current LLMs.