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You've come up with a brilliant idea, you spent time and money to implement it, and you want to make sure you receive the credit for it. I get that.You're thinking at the individual scope. When making a policy, a given individual doesn't weight much. Even all inventors put together don't have much moral weight, compared to the rest of the population.
So, patents rewards inventors. That's an advantage. But it's a little advantage, for there are not that many inventors to begin with. There are also the costs associated with the implementation of patents (patent bureau, patent lawyers…), which may be significant, but not much in my opinion. So, on first approximation, we can ignore those aspects, because there are bigger effects, that affect society as a whole.
On the disadvantage side, patents restricts what people can do, and they kill competition. (Of course. They were designed that way. But it's still a disadvantage.)
On the advantage side, patents are an incentive for inventors to invent, and for companies to fund them. The inventions then benefit everyone (starting with rich people, but still).
Now, all we have do do is measure which weight the most. See how patents actually affect society, and decide if it's a net benefit, or a net loss. Either way, try and think about policies which may have more advantages, or fewer drawbacks —or both. Patents are far from the only way of encouraging innovation, after all.
You can see that it's much more complex than the "reward the inventor" principle. Principles are good, but they're often not enough. When you look at actual consequences, you see that sticking to easy to understand principles is not always the best thing to do. Especially when deciding for an entire country.