You weren't boosting the dead car, you were powering it totally via some longer leads. The longer leads have losses, so shorter is better. You're right that thinner leads also have higher losses. So short&thick cables are best.
Once the car is started, it should be able to continue running off the output of the alternator even without a battery. Assuming the alternator and wiring is working correctly, etc.
The battery is required only for starting the car, else you'd need a crank handle or some other way to spin the cold engine over. Imagine a rope-pull on a car, like a lawn mower or chainsaw, or a kickstarter like a motorbike.
The other purpose of the battery is to provide power when demand is high. Sometimes the alternator cannot provide enough power to run all the lights, heater motor, electric fans, and options like heated seats or large audio amplifiers, and the battery helps bridge those load-spikes.
In theory you could drive the running car without a battery, but protect the positive battery lead so it doesn't short-circuit onto any part of the chassis. Also, don't stall!