Re: 'Noticing' smoke
A search for "redox battery" brings me to a Wikipedia article on flow batteries, which current EV batteries definitely aren't. If you want to be technical about it, all electrochemical cells perform "redox" reactions, which makes it trivially true but meaningless.
The important part of a Lithium-Ion battery here is the organic solvents in the electrolyte, which definitely need an external source of oxygen (or to satisfy pedants from the previous article's comments section, an oxidiser) to burn.
The reason EV battery fires are difficult to put (and keep) out is that once they're compromised, even if they've been "put out" they can continue to produce large amounts of heat even when they're not burning, which allows them to reignite for a significant period of time.