Ah, you mentioned two other problems of supercaps.
First, many of them have ridiculously high internal resistance. This means that they are not capable of delivering high current. You can short a supercap, but the current will be limited by its internal resistance. If you use the supercap as a battery, it will be heating up. Ideal use of a supercap is a backup energy source that is drawn upon only for a fraction of a second - to park heads of a HDD, for example.
Another problem is that they are low voltage parts. This seriously constrains your options because most semiconductors do not work well at very low voltages. You cannot trivially chain several supercaps either, not without special measures to protect individual cells from overvoltage or reverse polarity. Remember the failure of a Li-Ion battery on Dreamliners this winter? One cell out of many failed and ignited the rest.
Yet another problem is that they cost more than the rest of the phone. That one supercap, just 100F * 2.5V, only holds 648 joules of energy, even if you know how to take it out of the part. It's only 0.18 watt hours; probably good for talking for about 5 minutes. But the part costs $35 in quantity! A Li-ion battery would deliver 1A for an hour at 3.6V, resulting in 3.6 watt-hours, and it will cost ten percent of that.
Supercaps are good for what they are good for; but they are not a universal solution to every problem. If they were, they'd be already used for that. Engineers of Nokia, Motorola, Ericsson and of everyone else in mobile market are not exactly incompetents who don't read trade magazines.
I have a few supercaps here on my bench. I have no use for them, so special these devices are. I bought them to use as a "power loss" backup for an SSD that I was building, but the design already went past that concept.
I saw one article claiming that you could use them in your cell phone AND jump start you car with it. LOL