No. Look at the Cold War, especially the Cuban Crisis. There were a lot of voices on both sides openly calling for a first strike, and the cooler heads prevailed.

India and Pakistan are both nuclear armed and neither has pulled the trigger.

As for Putin, think of it this way. The people who told Vlad that the Ukraine invasion would be a walk in the park are the ones who would actually pull the trigger. Those folks were robbing the army blind to pay for dachas and harems; do you think they'll risk their Paris apartments and yachts to keep an old man happy?