It is unlikely but possible that suicides from the shutdown will exceed deaths from the disease. But it is absolutely not true that they will exceed projected deaths from letting the disease run its course unchecked.
Here is a parallel. We are afraid of buildings catching fire, so we build sprinkler systems into them. But the result is that in actual fires, the water damage from the sprinkler system exceeds the damage from the fire. That's because there is a fairly small fire and a big wet area from the sprinklers.
Is this an argument for not having the sprinkler system? Of course not! Because the damage from the sprinkler system is far less than what the damage from an unchecked fire would have been.
By my estimate, the deaths in the USA alone, with no lockdowns, is around 2 million this year. Other attempts that I’ve seen to estimate it come out with similar figures.