Think about your question for a minute.
Buy a Tesla. It has an 85kWh battery. Now add about 100kWh worth of these supercapacitors because battery charging isn't 100% efficient and neither is discharging supercapacitors. Why would you want to haul around both 85kWh of batteries and 100kWh of supercaps when you can just bring 100kWh of supercaps?
I see Supercaps are being more used for rapid discharges/re-gen periods. I.E. Tesla P100D (0-60 in 2.5sec) punching it would use the supercaps, and then going 100mph and letting regen take ahold, would also use the supercaps. The system would then balance out the power in the supercaps and what can be put into the battery for longer storage. Kind of like a crazy audio system. Has the supercaps that charge up for the bass hits.
You've correctly pointed out that the supercaps are too big and heavy and such. But let's just imagine for a moment that they're made of magic and are 5% of the volume and 5% of the weight of batteries. It's still not a good idea to charge them up to then charge the batteries (from fixed charging stations) because then they're better than the batteries full stop. Throw away the batteries and use them the end.
The only way that they make sense is for what you've described, they're electrical buffers to help impedance match "slow" batteries with "fast" motors. But since that wasn't what the parent was asking, I didn't answer that question.
They rarely get hot enough to melt, even though they pretty much absorb the entire braking power.
I'd calculate the resistance of copper cables vs car mass and delta v, but I'm too lazy.