Internal combustion engines produce power in short duration pulses. The rest of their cycle is devoted to preparing for the next short duration power pulse.
There are two ways to increase their instantaneous load capability; produce power pulses at a faster rate such that they are more “continuous” (e.g. more cylinders and/or higher rpm) or store the energy pulses in a large rotating mass such as a flywheel. The latter is not well suited to a modern automobile application.
Contrary to your thinking, there is not a lot of energy being stored in the rotating mass of the automobile engine. Responsiveness to power application or removal is highly desired in a car engine. It’s designed to have just enough rotating mass to keep it running smoothly at idle with a minimial load applied to it.
By gradually releasing the clutch you reduce the instantaneous load on the engine and by pressing the gas pedal, you increase the engine rpm and therefore produce more power at closer intervals. Both of which help to overcome the load being applied to the engine. There is an optimum range where the clutch can be released with minimal slippage and the engine rpm is capable of handling the instantaneous load of beginning to move the vehicle.
Feathering of the clutch and gas pedal is an acquired skill. The amount of feathering required is dependant on the vehicle weight, engine power, clutch design etc.