Does suffering motivate us to achieve our goals? Here's an interesting situation.

Let's say we have to persons. Tom and Happy. Both have the same desire: "I want a new car." The strength of their desires are equal. Tom lives in universe A, Happy lives in universe B. Both universes are exact copies of each other, everything is the same, timeline is the same, events are all the same, everything,.. Tom and Happy are copies of each other, clones, but there's just one little difference between the two universes: Tom suffers, Happy does not suffer.

Tom suffers: "I have an old car with bad engine and that's why I'm not happy. I want a new car."

Happy does not suffer: "I have an old car with bad engine and I'm happy. I want a new car."

Both persons want a new car for the same reasons: "Because it's old and has a bad engine."

The only difference between Tom and Happy is that Tom is suffering because of his old car with a bad engine; and Happy is not suffering because of his old car with bad engine. There are no other differences between Tom and Happy and the two universes they live in.

We can see that both Tom and Happy have the same motives for getting a new car: old car with bad engine.

If Tom and Happy have the same motives for getting a new car, both will exert the same amount of energy to get a new car. So, both will do the same exact actions in their universes to get a new car. So, both will get a new car at exactly the same time. 

But Tom has one motive more than Happy: Tom is suffering! Does that mean that Tom will get a new car quicker than Happy?! If Tom has more motives than Tom, than it's logical that Tom will be more proactive in getting a new car than Happy? So, the question is: who will get a new car quicker, Tom or Happy?

The real question is: is suffering a motivator? Is suffering a motivator only if the amount of suffering does not go over a specific threshold? Or is suffering not a motivator? If suffering is a motivator, than it does not make sense to completely remove it, because if you remove it, you will be less proactive to reach your goals in life? What did Buddha say above this? I'm a beginner and I'm sorry if I don't understand, I hope someone can shed some light on this topic. Thanks
In your example you stated that both Tom and Happy had cars that had bad engines.

Tom was unhappy and Happy was okay with it.

So right from the beginning we can see that their happiness did not depend on whether or not they had bad engines but their reaction to it.

Therefore one must apply the same conclusion to the outcome of getting a new car.It's not about who gets the new car first,it's their reaction to it.

Happiness depends on reactions to circumstances.Not circumstances themselves.

Is suffering a motivator?

Yes absolutely.When the Siddhatta saw suffering he decided to pursue the spiritual life.Sometimes living in deva like bliss masks the true nature of reality.That it is impermanent,not self,and suffering.

Is suffering a motivator only if the amount of suffering does not go over a specific threshold?

Yes that is why the human realm is the best place to practice the dhamma.One has their fair share of happiness and suffering.It is balanced to some extent.

If suffering is a motivator, than it does not make sense to completely remove it, because if you remove it, you will be less proactive to reach your goals in life?

That's like saying bleeding will motivate you to bandage yourself.But if you do bandage yourself,then the bleeding will stop,then how are you going to bandage yourself..?? 

Removing suffering is the goal.