It’s a quick motion, but the essence is that while moving from the brake to the gas, you’re also starting to apply the clutch to grab even at the still idle speed of the engine. It’s not several steps but a fluid motion, and as weird as it sounds, it’s something you pick up by feeling what the car needs to maintain the right engine speed while also not engaging the clutch too much and causing lugging or a stall. It’s why most new manual drives start in a empty level place like a parking lot and practice just going from stop to moving slowly, over and over. I also told both my boys the first time they got behind the wheel the same thing my dad did - you WILL stall out the first time. And they did. :D But they both have and love driving stick now, and hate if they have to drive someone’s automatic.
Oh yes, it’s absolutely dependent on the car’s abilities. But honestly there’s nothing wrong with using the handbrake if you need to in a bad situation. I’d say it’s a sign of a good driver to know the option is there and to use all of them together to get going safely.
I had to use the handbrake once on a 73 Beetle to prevent disaster. I was coming to a stop sign near home, pressed the brake, and it went to the floor. The sensor on the brake cylinder had shorted out and melted a hole, and the brake fluid went right out. Terrible design. But my awareness of what was left made me go straight to the handbrake and slowly come to a stop just in time, then I limped back home and figured out what had happened. So it’s not there just for a parking backup.