Yes, it is an inherently unsafe battery chemistry. And they're everywhere. I really hope we find a better replacement soon, there are already some promising contenders but none that meet-or-exceed on all specs.
That training was probably designed more to instill paranoia and cover legal arse, because my initial reaction upon reading that was "What the fuck!? No. No. No. Not even close."
Burning fluoropolymers will produce HF. Lion electrolyte is, besides being rather flammable, nowhere near as dangerous. It's a mild solvent. A leaking cell, if it's not burning, will not expose you to HF. In fact I'd say the risk of chemical burns from a leaking alkaline battery is higher.