You wouldn't need an equivalent capacity in supercaps as batteries, only as much as the vehicle can use or create in a short amount of time.
The UX would be invisible: If demand is greater than the supply available from the battery pack, drain some energy from the supercapacitors. If it's less, gradually shift some energy into them. If the driver is not in the vehicle, drain it all out, when they come back, recharge the supercaps.
The problem is that these capacitors and transfers add cost money, space, weight, energy, and complexity.
Looking at some numbers, batteries are ~20x more energy dense in both volume and weight. Batteries have 1/10th the power density, but efficiency drops significantly with discharge rate and they have fewer lifetime charge cycles and poorer temperature response. Batteries also have much, much lower self-discharge, and cost a lot less per kilowatt-hour.
Right now, lithium-ion batteries are simply good enough and supercapacitors are not that much better, so it's the cost-benefit analysis says it's better to take the costs you would have spent on supercaps and just put in a few more batteries.
That balance could shift, though, with improvements in supercap technology, or - paradoxically - with improvements in battery chemistry. If a battery with much better energy density and/or lower cost was found, with a reduced discharge rate that was sufficient to cruise but insufficient to accelerate or do regenerative braking, you could add supercaps to make a vehicle based on that battery feasible. It's similar to the electric grid - the base load would be handled by the slow batteries, similar to coal or nuclear power plants, while peaking loads would be handled by the supercapacitors, like a gas turbine.