The nukes wouldn't actually GO OFF round the clock, only when a kaiju emerges.
They weren't coming out every minute, in fact for years they were only appearing every few months or so. For all they know the invaders would have given up after losing 20 kaiju moments after exiting the breach.
If you want to act like the most sensible tactic wouldn't be to have automated nukes at the breach fine, I'm not to argue it. If you think the plot is plausible and every element makes tons of sense, more power to you. I enjoy the film greatly in spite of its "check your brain at the door" moments.