Making the pedals on my kid's bike rotate when I push so he'll learn to pedal

What you describe is fixed gear bicycle. Generally they are recommended only for people who are already very proficient at riding a bike, not kids under 3 years.

So your boy is simply sitting and not pedalling, so you push him along? Of course he's not going to pedal when he's getting a free ride from you.

Stop Pushing!

Firstly, does he understand how a bike works? Do you ride? Does he see you or others riding?

You might consider going the other way, and remove the whole transmission and the training wheels. Essentially revert the bike to a balance bike until he gets the idea that he has to push.

Something like this:

Eventually he'll realise how slow the balance bike is, and want his pedals back to go faster. And that's awesome.

Two options:

1. Remove the pedals (leave the crank in place) and set the seat low enough for him to touch the ground with his feet and push himself along. Keep raising the seat a little at a time, as he gets the concept of balance, and reinstall the pedals when he seems ready.
2. Intentionally do what is often unintentionally done to a bike: Wrap something -- rope, twine, some sort of rubber strip -- around the rear axle, between the wheel proper and the sprocket. Get it tight (tie it in place if necessary) and make sure it doesn't stick out to where it will get caught by the chain. This will keep the sprocket from "freewheeling".

As to #2, two cautions: 1) There is a very slight chance of serious injury if the kid gets going downhill with his feet off the pedals, then sticks his feet into the path of them. 2) There is also a slight chance that the bicycle will be damaged by the wrapping, and the forces between the two moving parts. (However, I've seen several bikes with this done unintentionally, and, aside from busted spokes on larger bikes, there has been no obvious damage.)

I have 2 kids, they both knew how to ride a bicycle around 2.5 years (with pedals and without training wheels).

The secret was to teach them only 1 thing at a time, in a funny way.

Basically, here was the plan :

1. Move forward,
2. Use the direction,
3. Balance,
4. Use of pedals,
5. Everything combined.

Balance bike

The bike they used the most during all this process was their balance bike. We bought one and give it to them around 1.5 year. Since then, we proposed them to use it every time it was possible (going to the park, at nurse's etc).

Basically, kids will learn steps 1 to 3 by themselves.

You know they rock at step 3 when they intentionally raise their feet just for fun in slopes, or just after strong pushes with their legs.

If they raise their feet, they're ready for step 4 : learning to use pedals.

Bike with training wheels

Make this step as short as possible, and make them alternatively use the balance bike.

That's not simple to get the idea that you need to push on on only one foot, and alternate. In terms of coordination. But as they know they need to go forward, just showing them once or twice will make them get the principle.

Once they get it, they just need practice to make it automatic. Then they're ready for real bikes

Without training wheels

Once you tried this, you can forget the training wheels : you still can alternate between a bike and a balance bike.

Find a slightly downhill portion of asphalt (you need it to be long enough, and safe). Place your child on a bike (without training wheels) and if he's ready, just help him place his feet on the pedals and run next to him as he will enjoy the ride ?

After a few descents, he will get it completely.

I had the same problem with my son. He didn't understand how to pedal and he lacked the strength in his legs to get it moving easily and so form any association with pushing and movement. I was advised to use a push-bike without pedals, but obviously this doesn't solve the actual problem, but rather just avoids it.

In the end, I jacked the back of the bike up on some bricks (on the stabiliser wheels) and then manually moved his feet around on the pedals until he had figured it out.

I also got him at first to just give little pushes on the pedals rather than a full 360 turn.

I had the same encounter with my 2 year old in 2015, then winter came. When the 2016 summer came ~8 months later, he at 3 years old understood it with no effort.

Just relax, and as his cognitive skills grow, he will learn quickly.