What is it that specifically kills ALS patients?
TL;DR Ventilation assistance does help but there are other factors.

ALS is a degeneration of the motor neurons, replacing them with hard material (i.e. sclerosis). The neurons innervating the muscles involved in respiration deteriorate and so it becomes difficult to breath, swallow, cough, etc. There's a compounding effect: we know that blood O2 and CO2 levels are balanced by respiration, O2 in/CO2 out, and we know that swallowing/coughing clear secretions from the respiratory tract. Basically you end up with trouble oxygenating your blood and riding it of CO2, hence CO2 poisoning. This can result in what is known as type 2 respiratory failure, characterized by low oxygen and high carbon dioxide levels (hypoxemia and hypercapnia). Low secretion clearance in the lungs can also result in aspiration pneumonia in addition to poor gas exchange.

Ventilation methods such as BCV are capable of mechanically assisting breathing, and maintaining ventilation-perfusion state in a way that prevents respiratory failure, but unfortunately doesn't slow the progression of the disease. The problem then is that in, for example, particularly severe cases, the quality of life has to be taken into account because tongue movement, gross movement in general due to motor neuron degeneration just fails. This can lead to what is essentially locked-in syndrome (LIS), and clinical outcomes can widely vary, though in this article they report the more pleasant outcomes of patients experiencing LIS. I think that for these cases due to poor communicaton patient-reported outcomes are fairly underreported, but family and caregivers at this point have to take into serious consideration options such as euthanasia. Cutting off ventilation can result in type 1 or 2 respiratory failure.

Now, I lack actual data for this statement but I should also point out that not all patients receive the same care or opt for assisted ventilation, in which case late-stage ALS would invariably lead to total neurodegeneration to breathing muscle innervation, where assisted ventilation becomes a requirement for survival.