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Markets only function well to some extent. Just having a market does not mean that you have a system that solves all pricing problems.

Your assuming markets are perfect - they aren’t

For each product needing to be rated, there's one seller and potentially many buyers.

Sellers need to convince buyers to buy their product. Sellers need to get their product rated in order to appeal to buyers. It's a one time cost to the seller to appeal to a large amount of buyers, and the higher the rating, the larger the pool of potential buyers.

Flip it around. How do you coordinate enough buyers together to pay for rating all the potential products out there to buy?

Yes, you are right. Also because rating can't be kept private, so there is no reason for any buyer to subscribe and then rating agency isn't viable.

As ajross says, there is a conflict of interest here. Not everyone wants an unbiased rating.

2008 provided strong empirical evidence that the market does not have the motive to solve its own problems. Markets are not obliged to conform to theories about how markets should work.