The plan was to charge and go, but it turned out that, after connecting to the Supercharger, the car showed charging at 0 kW. It's not a good sign, as the car uses a lot of energy for heating the cabin and it's -28°C outside, so you have to act quickly.
Well, the unaware driver switched to another stall thinking - maybe this will work - but it didn't. So he switched to another one, and another one... all of them, trying to find one that would actually work. But either the Superchargers were not charging (error) or charging at 0 kW.
***UPDATE: DC charging a Tesla doesn't display the amount of energy being sent from the Supercharger to the vehicle for heating purposes. Indeed the car was likely drawing energy from each stall, but the owner of the vehicle would've had to wait awhile for charging to start.
Since the support call on Sunday turned out to be useless and the SOC lowered to 10%, it was time for plan B: find a Level 2 AC charging point - in this case, a Tesla Destination charging point, 10 km (6 miles) away.
Well, it took a while, and the SOC dropped to just 2%, which is pretty risky in such circumstance, but at least the AC charging worked immediately - several kW is not much, but enough to not be stranded.
***UPDATE: It's likely the brief drive to the AC charger warmed the battery enough to accept some low level of charge.
While most of us know what the problem is, it's very important for all the new EV drivers, and future EV drivers, to know the specifics of cold weather charging.
The root cause of the problem is not the outside temperature of -30°C, low state-of-charge or a broken Supercharging station. The root cause is the cold battery and its limitation (in most of the lithium-ion chemistries) of charging at low temperatures.
Internal combustion engines have their own problem with starting in cold weather. EVs, on the other hand, might not charge when the battery is cold.
The best practice is to keep the battery at a reasonable temperature, through several actions. First to pre-condition the car (cabin and battery if possible) before starting the journey, using energy from the grid.
The battery gets warmer while driving (due to discharging and/or thermal management system), but to make sure it's warm enough to fast charge, it's worth pre-conditioning it ahead of the upcoming fast charging on the route - Tesla specifically has such a feature in its Navigation, which points to the Supercharger and heats up the battery).
The software probably has some thresholds to keep an energy reserve, so the pre-conditioning might not be possible while driving at a low state-of-charge. This is why it would be smart to set pre-conditioning early and/or plan to arrive at the fast chargers at a relatively high level (like 25-30%). TESLAPEG's video indicates 20% or more because 11% was not enough.
Even then, you must be prepared for Supercharging power output to be limited. This will lengthen the charging session.
Finally, AC charging is a very reliable option - it was able to heat/charge the cold battery, because its power is relatively low compared to the battery capacity - like 0.05-0.1C rate. We guess that the DC fast chargers struggle to deliver such low power and this is why they were not starting even at a few kW.