It takes a lot of power to heat the batteries. If there is no power, can't heat them, and you can have total failure if you try and charge them when they are too cold.
You don't get total failure, you get an extended charging session as the car first warms the battery and then charges the car.
The car needs enough power to warm the battery though. So what happens if the car doesn't have enough power and tries to warm the battery and runs out of power before it has warmed enough to charge?
Why would you need to use car power to warm the battery? If you're connected to a charger you can use charger power to warm the battery. And you can't charge if you're not connected to a charger, so charging a cold battery is impossible in that case.
There should be a lot of power available if you're at a charger.
My understanding is that this particular Supercharger station is broken and either not putting out any electricity, or putting out way less than it should
I think battery heaters are standard and necessary on any EV battery.
not all electric vehicles have battery heaters.
Eg it's only part of the Eco pack for an Ioniq.
The issue at hand may be whether they keep the heaters on during charging. FTA: "A charge that should take 45 minutes is taking two hours."
They do keep the heaters during charing, unlike other manufacturers like Renault