True but you've got to consider the numbers we're looking at here.
As an example an iPhone 5 has a 5.45Wh battery, if you wanted to replace that with a super-cap and charge it to 5V in 20 seconds you'd need to provide ~1000W of power or 200A @ 5V. Even if the super cap had very low ESR, call it 1mOhm, which is extremely low compared to current super-caps, you'd still end up dissipating 40W as heat with ~96% efficiency.
Perhaps phone makers can start making their cases out of aluminum, with enough mass to take the heat. 40W over 20 seconds doesn't sound too daunting.
If you split such an aluminum backplate into two parts you could use them as the power contacts. Then you could have a "coffin" type charger where you put in the phone, then closed a cover to run the charge cycle. Kind of like the cover of a washing machine, to reduce the danger level.
It would be 40W _for_ 20 seconds, or 800 Joules, which is a lot of heat, enough to boil 3 grams of water that started at 20C or increase the temperature of 30g of aluminum by ~30C over ambient.
14AWG (the small one) is the one rated for 20A which you might find in the power cord for a desktop PC, and is significantly bigger than the wire on your current phone charger. The big one (1/0) is rated for 125 amps. You have to go to 3/0, two sizes higher than that, for 200 amps. 3/0 gauge wire is what they commonly use for the main electrical service for a commercial building.
There is no way they would use a 5V charger if it had to draw that much current. But then you have a different problem: High voltage DC is extremely dangerous because it causes your muscles to contract if you come into contact with it, so your heart stops and you can't move to separate yourself from the electrical source.
20 seconds for a full charge is just unrealistic. Make it 60 seconds, and use a 24V charger, and now you're well within reason.
Your right that the way to solve the current issue is to increase voltage, but that creates a list of other problems, the most notable of which, death, you've already touched on, but here some others:
1) You likely still want USB charging support, so now you need a boost converter in the phone or cable.
2) Your PMIC needs to accept the higher input voltage, and you have to be willing to accept the reduced regulator efficiency from the increase in Vin - Vout.
3) 24V running around in a phone creates a lot of possible problems that you don't have with 3.7V cells. Increased moisture sensitivity, gradient induced oxidation, etc.
4) You still need a ton of power. If you jump to 24V then you still need ~42A to hit a 20 second charge, go to 60 seconds and you need ~327W or 13A @ 24V. That is still a massive charger with 10 gauge wiring (1 conductor is 2x + the diameter of the entire lightning cable).
Keep in mind these are lower bound numbers all around. Reality could be 50-100% higher for power needs. The ESR value I threw out above is also a very optimistic minimum hoping that this new tech has much better ESR characteristics than current super-caps which for large capacity models can be up in the hundreds of mOhms which causes a huge thermal issue.
Its still a long ways from being even remotely reasonable for super caps to replace batteries in high power devices like phones and tablets.
As an example an iPhone 5 has a 5.45Wh battery, if you wanted to replace that with a super-cap and charge it to 5V in 20 seconds you'd need to provide ~1000W of power or 200A @ 5V. Even if the super cap had very low ESR, call it 1mOhm, which is extremely low compared to current super-caps, you'd still end up dissipating 40W as heat with ~96% efficiency.