The peak power at work when a flash is being discharged is extreme.

Example: a classic Metz 45CT (a large but still handheld unit) on full manual power delivers around 90 watt-seconds of electrical output to the flash tube in 1/300s. That means there are 27000 watts at work for a short time.

A 27000 watt lightbulb (BTW, your flash is much brighter than that - more efficient!) would need the ampacity of around eight european household sockets (230V, fused at 13 or 16 amps) to operate, since it would draw around 120 ampere from them. At the voltage levels xenon flash tubes typically operate at, around 80 amperes will flow; if you wanted to pull that energy "live" from a 5-6V battery, you'd need to pull around 5000 amperes AND instantaneously convert them to around 80 amperes at high voltage. Batteries that could deliver 5000 amps - while not wasting most of the power as heat generated across their internal resistance - for a short duration and live certainly exist - you are looking at starter batteries for big lorries, which are rather inconvenient to lug around. Also, a 5000 ampere current pulse is quite a magnetic nuisance if there is anything either ferromagnetic or intended as an electrical conductor near the conductors used.

Your average "big" speedlite, btw, will be around half the power described here (but might in some cases reach the same or higher peak currents in order to discharge even faster), an in-camera flash will be a fraction of these numbers - but still inconvenient to build any other way.

If you need a short, massive current pulse, the best energy source available in engineering is a previously charged capacitor (metal-paper or electrolytic type, not dual-layer "supercap"). This energy storage device is large for its storage capacity if you compare it to a battery - but able to respond far quicker to high current demand, since a) there is no chemical energy conversion involved in its operation, b) the relevant current path only involves metallic, very conductive parts instead of ionic conductors (aka electrolytes. An electrolytic capacitor, btw, does use its electrolyte for chemical self-upkeep, not for chemical energy storage or electrical connectivity).

Also, since the charge in a a capacitor is limited, it allows portioning total energy to what the flashlamp can safely take. This allows a flash in manual mode to just completely discharge the capacitor until it is nearly empty (when the remaining charge can only deliver voltage below the quench voltage of the flashlamp) without complex control circuitry - and at the same time gives you a safety fallback when working with dynamically controlled power levels (telecomputer or TTL mode), leaving no way for catastrophic circuit failure in combination with several kilowatts of constant power available turning into a massive safety hazard. A handheld-sized efficient electrical near point light source with 27000W input, driven constantly, would first severely damage your eyes and then self destroy violently (think of a welding arc - dangerously bright, and it melts metal).