kid's bike rear brake not having stopping power

This is a single pivot sidepull brake.

They have a flaw in the centering spring that makes the brakes center unevenly. Because of this flaw, single pivot sidepull brakes are made with low mechanical advantage so that the brake pads have enough room to retract, in an uneven manner.

The best fix for your problem would be to find a single pivot sidepull brake that doesn't have the flaw in the centering spring. An optimal centering spring would have the loop not in mid-air but rather around the brake pivot bolt. Presumably such a single pivot sidepull brake would have high enough mechanical advantage that reasonable finger force would stop the bicycle.

Unfortunately, such brakes are not available. The manufacturers of sidepull brakes do not realize the problem and instead came up with a different, poor solution: forced centering. This forced centering is present in dual pivot sidepull brakes. They have several problems:

• They cannot track a wobbly wheel
• The brake pad on the offset pivot arm goes up as the pad wears, eventually touching the tyre and destroying it, unless the pad is very thin it wears away before this happens, or unless the pad position is adjusted all the time

However, dual pivot sidepulls have reasonable mechanical advantage so they can stop the bicycle just fine.

Your brake looks like a very long reach brake so I'm not sure if dual pivot sidepulls are available with such a huge reach. You may be stuck with the poor brakes unless you replace the entire bicycle. If selecting a bicycle that has room for large tires, you should prefer V brakes and cantilever brakes over sidepulls as V brakes and cantilever brakes don't have the flaw in the centering springs so they can have high mechanical advantage and reliable centering at the same time.

Sadly braking at the rear wheel is relatively poor because of a combination of factors.

1. Kid's hand strength - your average 4 year old can't squeeze the lever as hard as a larger rider. This affects braking at both wheels.
2. The brake lever's pull ratio is also often sized for a full sized brake so it has to move through a certain angle to pull enough cable to actuate the brake. For a little hand the lever can be scaled down, but it still has to pull enough cable, and the brake calipers are not scaled down. Notice you didn't have to fit a "kid sized" caliper.
3. Mass - the rear wheel's brake is never as effective as the front wheel's brake because the system's center of mass is higher than the axles. As the brake bites, the rear wheel becomes unweighted, so is more likely to skid.
4. Cable friction and a longer run of outer - You've already addressed these.

You will generally find the small 12" wheel bikes to have a backpedal or coaster brake because kids have better leg power than hand power.

As such, kids can do "mad-skidz" with a coaster brake but not a hand brake. Part of this is the body motion required to stomp on the rear pedal naturally weights the rear of the bike, while I've never seen a kid consciously do the same while hand braking.

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You might try roughing the braking surface of the rim very lightly with sandpaper.

As an adult you should be able lock the wheels easily while wheeling the bike around while you walk beside.

Another test option is for you to try riding the bike and test the brake function. If its really way small, you can sit over it and keep your feet on the ground rather than pedalling.

Do basic checks, are the brake levers bottoming out on the handgrips? Are the wheels true ? Can you take some of the gap out so the brake pad sits closer to the rim, but does not rub.

Good luck ! And remember, they grow up fast and will be on a bigger bike in short order.