Summary: Performance reviews are a key part of management and only managers are really qualified for it. Gamifying the review process is a terrible idea.
Your concerns are very valid. Even if the potential problems you mention, like people gaming the system, don't actually take place, the mere belief that they could be present is poison for morale.
Suppose you've got nothing but star performers and all of them get great feedback. Some employees can be more likeable than others or, as you say, do more visible work and will thus get more recognition. Management has already locked itself into a set number of "rewards" which means that a lot of great employees will feel left out. The potential problems of this are numerous:
- employees feeling unappreciated
- resentment for people who got rewards who didn't deserve them as much (even when they definitely did!)
- employees can't accurately judge each other's work or responsibilities (this is part of the problem for the previous point)
- promotes the belief that only the star players can build their career
- managers replacing honest feedback and close monitoring of performance with a gimmick like this
And there are probably any number of things that don't even occur to me. The bottom line is that there are so, so many downsides to this problem that they vastly outweigh any perceived benefits, especially long-term. I'm a strong believer in rewarding performance, but that is an essential part of a manager's duties and shouldn't be turned into a game. And this is assuming we're dealing with a team of high performers. In reality every team will have people who are merely adequate, struggle or who aren't performing at the level that the position demands. These people should be managed, not be indirectly and invisibly punished by such a reward program.
So yes, you should definitely give your feedback on this. You've already articulated your points well in your original question and you're free to borrow some of the issues I've raised to argue your case. I'd suggest doing a further online search on this topic as there are dozens of blogs and articles on this topic and it's possible that there's actual research as well that could also strengthen your point.
One thing that I always recommend doing in a situation like this is to argue against this as a group. Many of your colleagues will share your apprehension and it's always better and more efficient to raise things like this as a group.
One final note and something that I recommend you mention is that their original plan is actually a good one. It's rewarding and useful to have colleagues give feedback on each other or recognise their achievements. Whether this recognition should be passed on, passed on anonymously or reserved for management can be debated but it's a great concept. It should definitely be optional but it's a great idea to encourage people to think about it before or during a review. It's attaching "rewards" that's the real problem here.