There's very little free lithium in lithium-ion batteries - it's usually largely bound (via intercalation) to either the kathode or the anode.

The stuff that starts the fire is the electrolyte - ethylene carbonate.

What else ? If they are burning it's already too late to stop it, you're just cooling it down and reducing damage to the surrounding at that point.

That's the problem with extinguishing batteries, you're not, you're just limiting the damage.