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High drain isn't the problem, most lithium ion can be drained at 15-50C (drain rate not temperature) until empty without being harmed.

P100D is 100kw/h battery, which should be able to put out at least 1500 hp continuously.

The problem is 100% Tesla's cooling system not being powerful enough.

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Not all lithium ion can be drained at high rates like that. There is a trade off of safety, lifetime, energy density and power density (discharge rate). For instance the RC helicopter batteries are rated for 100C+, but are notorious for fires and short life time. As it happens, the RC photo drones are easier on the batteries and want longer run times so there are batteries that have higher energy density but can only handle 20C. A lot of e-bike packs are built from 1650 batteries and these are best off when limited to 3C. I'd imagine the priorities of a car battery are similar.

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