The short version is that life needs something that's at least a little unstable in order to extract chemical energy from things.

The post is correct when viewed in a particular light, on a technicality, if you squint. By that same technicality iron rusting is also burning very slowly. They're ignoring the rapidity which is implied by "burning". But yes, oxygen is unstable, oxygen helps burn things, and oxygen is toxic if you get too much at once. Though you'd need to be breathing pure oxygen pressurized to about 1.4 atmospheres, or regular air pressurized to about 7 atmospheres, for that last one to happen. It's a legitimate concern for deep SCUBA divers.

But why does life need instability? Chemical instability is, in basic terms, just stored chemical energy, and that energy wants to be released. The more reactive something is the easier it is to get energy from reactions involving it. There's a balancing act here where more reactive means easier energy, but also more dangerous. Oxygen is in a kind of sweet spot where it's stable enough that it's not generally going to explode or catch fire on its own, but can be coaxed into doing those things in controlled ways with other chemicals to extract energy when needed.