That's not an assumption, it's a fact. There are a finite number of arrangements of protons and neutrons which can yield atoms. Each sequentially increasing count of protons is a new element. Adding neutrons to a given count of protons makes an isotope. We know which of these can exist in nature, and which can't due to their short half-lives / instability.

The quote doesn't assume you need carbon, the quote says that carbon is much better at the sort of chemical processes we associate with life than anything else. Nobody thinks all life absolutely must be made out of carbon.

But if you're looking for life, it seems very likely that most life is made out of carbon; possibly nearly all of it. You might as well start looking there. We have limited resources, after all.

> only elements we know about exist

The interesting thing with elements is that they describe almost all conventional matter in a very predictable pattern and we can pretty much enumerate them. Some scientists are even trying to make the next elements (Ununennium for instance), even though there are incredibly unlikely to occur in nature.

Granted, elements only describe a fraction of all matter, but we do not really expect anything complex to last long as plasma (stars), and we do not know much of dark matter anyway.

> no matter what conditions might be carbon is an absolute must for life

GP was answering this exact point. In short, it's just that the other elements do not look as promising as carbon for building complex molecules (whether we know them or not).