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.
