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The basic problem of the lead-acid battery is not voltage drop. In fact, on most cars voltage drops under cranking towards 9 volts, and truck industry rate the CCAs as the amps a battery can give under freezing cold while dropping to 7.2 volts. Neither is the (renowned) sluggish charging. Or water loss. Or limited life.
The basic problem which is unsolvable as long as we stay on this side of the laws of physics is weight. Lead-acid does not allow deep discharge below a certain threshhold, and in flooded lead acid this threshhold stays up, no more than 30% frequent discharge. Which means that we carry a battery 3 times as heavy as needed. Only to achieve a reasonable service life out of it. AGM, which is deep cycle by construction, for the same amps is even heavier. Which negates any advantage. Plus, even if by some magic we can size the battery to the exact amount needed to start the car and no more, lead is still heavy. As, well.. lead.
Supercaps alone are only a thread above useless. They can give a start, but they self-discharge in time, store miserable amounts of energy, discharge dangerously when some metal tools hits them inadvertently.
But once lashed to a battery, they can draw, store and fire the exact amount of current needed for a start - while allowing a small, lightweight battery to store energy needed for the car's systems. A 60Ah lead-acid is 15-18kg (33-39lbs), the lightest batteries rated for road cars are in the 10-11 kg (22-24 lbs) range, highly specialized racing batteries like Odyssey or Varley are a bit below 20 lbs. A combo of supercaps and motorcycle lead batteries is barely above 10-11 lbs and can start just fine.
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