Capacitors have self-discharge.
Couple that with the fact that the caps don't have a lot of amp-hour capacity, and your car will undoubtedly pull a tiny trickle of current to keep your clocks and settings in order...
So unless you plan to charge the car after it sits a few hours (which will undoubtedly result in a totally dead capacitor bank), you'll need a battery.
And then you've defeated everything.
Neat idea though as a boost option. If you could get 6 seconds of cranking out of a bank of them, it would be nice to have in winter.
2000+ cranking amps for the win
Edit: Also without inter-cell balancing, a single cell could (theoretically) be overcharged by the cars alternator, resulting in a nasty explosion (or just the cap venting, YMMV).