Good point, that is what a capacitor does when it fails, and then they explode or something weaker in the circuit will fail causing more and harder to troubleshoot problems.
Super caps as a battery replacement is not doable, never will be, rechargable batteries already fill that void and that's what they should be focused on.
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Supercaps are already being used on a small scale for battery replacement. There are several types of RC toys that use them (and are quickly recharged from dry cells). Also, I suspect that those "shake and use" LED flashlights employ supercaps.
There is no question that a large-capacity supercap (without internal limiting mechanisms) is a distinct safety hazard. But, the same thing applies to similar-sized batteries. (Remember the flaming laptops...?)
If I understand the article correctly, the plan is to pair supercaps with batteries. The supercap would accept the quick charge, and then would depete its charge by "trickle-recharging" the battery to spread the use out over a long time scale.
The problem of a high peak-current charging source for the supercap remains, however...
Probably a strategy like used in the toys would work: Supercap charging stations that are "trickle-charged" from the grid and that then can quickly dump part of that charge into the mobile power-supply supercap should be workable. That would reduce the peak load on the grid, while taking advantage of the fast-recharge properties of the mobile supercap/battery pack...
After a career in high tech, (semiconductors and micromechanics) I have learned that "...is not doable, never will be" is frequently a position that will come back and bite you in the rear... ( I remember something like that being said about superconductivity, for example...)