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Amazon removes reviews that call this out, so they do care — just in the other direction.

I just had an argument about this with Amazon support. They took down a review for an item where I mentioned that the manufacturer included a card in the item's box offering to pay you for reviews. Apparently my review violated their "advertising" policy

I had a similar experience recently because I called out that the name brand webcam was not in retail packaging. That was the last electronics purchase I’ll likely ever make from Amazon. I ended up returning it and actually found one at Best Buy a few days later.

My favorite rejected review was for 6 slate coasters that arrived as slate chips because they were shipped in a large envelope with no padding.

I think that might be unprovable.

A concurrent could always buy your product and send itself a fake mail promising a discount for a good review, then report the fake mail to amazon.

I don't know if that actually happens in practice.

better reviews => more sales => more profit

Do they? If a product has bad reviews, the person will buy an alternative... but likely also on Amazon.