Why aren't we ambidextrous?
Having a dominant side is advantageous. Instead of seeing it as the brain influences our fine motor skill, think of it the other way round. The way we use our hands or any other part of our body sends feedback to the brain to increase the portion of the brain devoted to that region of the body and of course with disuse the reverse occurs. The more you do a certain task the more that portion of the brain gets better and coordinating the movement. The portion I talk about is the primary and supplementary motor cortices. So basically we are specialising. In the same way a heart doctor is better with treating conditions of the heart than a foot doctor. So we can either devote our experience to the one hand, or both or the other hand. If we use both hands equally our brain adapts so that we are ambidextrous, if instead we use just the one hand our brain develops just that side. But if you do focus on both you are using your dominant hand less so overall same amount of development just over both sides of the brain rather than just one side. You can probably see the benefit of having this plasticity of improving only what is used because why improve what we don't use and waste resources? And you may also see the benefits of higher specialising of one hand rather than having two slightly specialised hands. There are few tasks that require both hands to be very specialised, but if that were the case then the body would go ahead and do that but in spending time training your hands you are probably training your legs or something else less.

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Why not have a switch between the two hemispheres to give the hand currently being used the fine motor skill located in the other hemisphere? (from comment)

I see. In my answer I only talked about one aspect. The brain doesn't have to learn skills for each hand completely independently. For example imagine if you learn to play a musical piece with your right hand only. If you try it with your left you won't be amazing but you'll definitely be better than a complete novice. Our brain can learn how to do by seeing which is vital for development. But there's a limit to that, your hands aren't completely identical so there's still a portion that needs fine tuning. Your muscles have different stregth, fatigue, resistance, recruitment patterns, places they innervate. Your hands are mirror images. The weight of your hand is different. And not only that but these variables aren't fixed, there's lots of them and they are different for different tasks. So to use your analogy, the more expensive remote can be used but it wasn't bought for use with the black plane. It'll give more control to the black plane but it still won't be the best ever until it learns what the muscle strength is etc. then it will be just as great.