Yes, Trident was designed to be a dead man's hand - there was always a second strike ability even if we got nuked first or were otherwise compromised.
However, a second strike ability is not the be-all and end-all of nuclear policy. It's not relevant when you have, as we have seen, a Russian state that is continually probing the West for weaknesses to advance its own gains.
By removing the threat of a first strike, you are telling your opponent that they can get away with whatever they want other than nuking you. If they march into Eastern Europe, they know as long as they don't use nukes you won't either. So you have a bloody and futile land war in Europe. But then the Russians say "if you don't stand down, I will nuke you."
And then everyone either dies, or you capitulate.
If you have a first-strike policy, the bloody war never happens and the brinksmanship does not happen. If Russia walks into Estonia, the world ends. End of discussion. It does not matter how many nukes Russia has. There is guaranteed escalation and guaranteed red lines. Nukes deter conventional warfare.