I would envision the main application of an ultra capacitor is the quick capture and release of energy on braking and acceleration. So, instant regen, and instant acceleration for short bursts, thus removing the load from the battery for longer term energy needs and less thermal effects.
I think you can really see the use of regen application in this vid of the Porsche 919 setting the record at the Nurburgring. It doesn’t actually use an ultra capacitor, but an 800v li-ion setup, where braking regen provides 60% of the energy, and a second turbo in the exhaust drives a generator providing 40% of the energy on acceleration and down the straights. Super cool setup. Watch the “boost” graph, which is the electric boost and regen. It happens really fast. The downforce of this car must be phenomenal.