Not quite correct.You know fire triangle, Fuel, Heat, Oxygen. Deny one no fire. So plenty of fuel is available, and heat is the problem so deny Oxygen.
Yes, once the electrolyte catches fire, Oxygen in the air is consumed, but Oxygen in a Lithium ion battery fire also comes from within the battery cells, specifically in the Nickel-oxide layers in the cathode (at least in NMC and NCA chemistries).
These cathode crystals exothermically decompose in a fire, releasing more heat & oxygen.
In fact, that's usually where the battery fires get started .. inside the cell.
In contrast, LFP cells (Lithium Iron Phosphate) have much more stable Iron Phosphate cathode crystals that release much less heat & oxygen when subjected to a fire.
That's a big reason why LFP cells are MUCH safer than NMC and NCA.
"The Limiting Factor" YouTube channel has a series of very informative videos with excellent explanations.