Handbrake first, let car rest, then set in first gear. If you do it the other way around, you're resting the gears under load.

Why is that bad? The gears surely take orders of magnitude more load in a moving car.

The real problem is you're not loading (and testing) the brake before putting the engine in gear.

An off engine is not unable to turn. Its resistance to turning is not constant either. All piston rings leak, even when new. So while you may have observed the engine in first gear holding the vehicle, minutes later the leaking piston rings may have allowed the crank to turn, and vehicle to roll.

The primary holding mechanism should always be the brake, and you need to load it to verify it's functioning and that you've pulled hard enough on the parking brake.

Putting it in gear is just a weak and largely ineffective form of redundancy.

Chocking turned wheels against the curb is much better redundancy (and legally required when parking on hills around here).

Though the engine won't stop it rolling away on a steep hill, it will keep the speed down instead of wildly accelerating. So that can reduce the danger and destruction it causes.

Once you are off the slope it will stop the car as well

Very well summarized

I always imagined it was a sustained load on a few teeth of the gear instead of constantly distributed as the gears turn.

There's basically no compression when an engine isn't running at least at hundreds of RPMs.

It's a non-issue for the gears. You can turn a 4-cylinder crank with your bare hands. There's very little load, you're just overcoming the static friction of bearings, piston rings, and compressing valve springs. At the static to creeping rate we're talking, the cylinder fluids leak right through the piston ring gaps.

If you want to worry about something with regard to statically loading everything, it's a lack of lubrication that should worry you.

The engine isn't maintaining oil pressure when stopped. So you're leaving the crank bearings loaded in one spot while oil slowly squeezes out. "Frictionless" oil film bearings are made of relatively soft metals, so this isn't exactly good for the engine.

Also the gearbox isn't dispersing oil as it does when driving. So the gears are slowly drying up while loaded. I'd be more concerned about the engine bearings though.

I never leave my manual vehicles loaded in gear when off. It's pointlessly hard on the bearings. Starting is already harsh enough in that brief moment before oil pressure builds. Why add insult to injury?

There are oil pressure accumulators on the market in part for maintaining oil pressure when off for friendlier starting of high compression race engines:

https://www.cantonracingproducts.com/accusump

I wonder what would be the load compared to a speeding car. Lets say car is standing on 10% incline, first gear, something like three teeth engaged. What is the load to one teeth? Now lets say you have started the car and accelerate it to something like 30km/h in this incline before changing to second gear. What was the load to one tooth in the time of acceleration?

I don't think that's a huge concern. Gear teeth take a ton of abuse from imperfect shifting. I've parked like that for 10 years in a car and first has felt the same as it always has.