Specifically where nuclear strikes are concerned, a lot of planning has gone into maximising the probability of compliance. Missile combat crew carry a sidearm to threaten their counterpart with death if they refuse to act on an authenticated launch order. They wear 4 point harnesses during launch sequence, primarily as a psychological device to prevent them from getting up and walking away mid sequence if their conscience gets the better of them, etc.
If trump or any other president said launch, there would be a launch.
https://en.wikipedia.org/wiki/Single_Integrated_Operational_...
Everyone involved in the chain is part of the Personnel Reliability Program. It's mostly to make sure someone doesn't launch without an order but also probably to make sure that they will follow orders, including launch orders.
https://en.wikipedia.org/wiki/Personnel_Reliability_Program?...
Ultimately if the President orders the launch, it will happen. And it's the President's decision alone.
https://www.bloomberg.com/politics/graphics/2016-nuclear-wea...
What seems to be his opinion is the danger of giving so much power in the hands of only one man (in this case the John Kennedy as the President of the USA and Nikita Khrushchev as First Secretary of the Communist Party of the Soviet Union). Never before the nuclear weapons one human being has had such a destructive power. In his other podcasts, he alerts about this danger in the current political environment as Obama worked to concentrate even more power in the presidency during his time and we never know who might end up being president in the future, it is a very risky thing to do. For some, Trump is the personification of this exact risk; but for others, who support Trump now, it worth thinking that they don't know either who might end up next at the presidency with all this power.