The reasoning is terrible.
The obvious solution to life being terrible is to take your own life. The reason given for not doing so is that death is somehow worse than remaining alive. How is death worse? The pain that usually accompanies death can be eliminated, so that’s not it. There only reason remaining for how death could be worse than remaining alive is that death keeps you from being alive.
This fact directly speaks to the worth of being alive. So you can’t draw the conclusion that being alive is bad, since the vast majority of us choose to remain alive. The real question is: how great is it to be alive? Here I think the philosopher engages in vast underestimation. To him, life is all misery and therefore not very worthwhile. But this is a ridiculous conclusion.
On average, human misery and suffering is at its lowest point now than at any point in our history. Our history is filled with war, disease, starvation, slavery, etc. The fact that we are here, that our ancestors did not choose to give up in the face of all this suffering, is not evidence that life is terrible. No, it is the opposite: it is evidence that being alive is such an amazing gift that they were willing to endure anything to remain so.
If you consider American slaves and Jewish holocaust victims, who suffered some of the worst atrocities in human history, as examples of people for whom the trade off between staying alive and ending their suffering with death may have been a fairly close call, then that means that, for anybody not facing those circumstances, the scales are tipped pretty heavily toward being alive being way better than not being alive.