That is an obvious solution for extremely expensive items that do not sell much. It doesn't work if the seller can just pay people to buy the thing and then write a review, no?

I think they should be able to tell apart from people's purchasing history if they are "organic" buyers [normal people's consumption patterns of goods] or something organized to curry favorable reviews.

Sure someone could get paid to review an item and also be an avid consumer of a product, but in that case it would seem they believe in the item.

Imagine someone buys eneloop over and over, never offer a review, then purchases a no-name brand, gives rave reviews and goes back to buying eneloop again.

Absolutely and recurring customers with a certain gained "reputation" should have more weight than new customers without much buying history. Maybe require a reviewer amass x-amount in previous purchases before allowing them to review as well as t-time before they can review to avoid fraudulent reviews.

I constantly get asked to review my purchases and have yet to do so. Some people are just not into doing reviews like that, specially if it feels like you're getting badgered.