A distributed trust network is a really interesting hard problem but does seem solvable in some ways (as seen by blockchains, though they have their own issues).

I've wondered if a somewhat simplistic graph based solution could work, inspired by organizations like medieval guilds and the mafia, where you generally need to be invited to the trust network and promoted to be more trusted within the network as others vouch for your trustworthiness. Suspicious users are highly punished (in the real life examples, often violently) and quickly removed from the network.

Obviously can be gamed like any other system, but would be much harder to do so and you can leverage the vouch/invite graph to detect trusted users that are highly connected to suspicious ones.

Surely there's something like this out there, though? Or is this just not a valuable business problem to solve?

It sounds like you're describing a product review site implemented as a distributed trust network, and the plan to establish the trustworthiness of contributors is to bolt on a contributor review site implemented as a distributed trust network.

maybe this works if you have to 1) be invited and 2) also pay to get in?

How many people in a "friends" list are people you actually know?

I want a review site where I can only see reviews from people I have vetted (could be pseudonymous).

Blockchain!

Proving that reviews haven't been removed might actually make this a real application for a blockchain.

Of course, there's no reason whatsoever to use a blockchain for this. The review website could simply offer dumps of their data signed with their private key.

Problem is, you will be forced by courts to remove some reviews for the usual reasons, making you instantly untrustworthy

You can publish data dumps and the internet will do the rest. I'm sure there are more than enough people out there who might either not care about the law or are located outside of that jurisdiction who would be more than happy to call out a scummy company when their existing bad reviews suddenly disappear from the latest data dumps.

You usually build a chain of hashes of messages, not of the messages themselves. Meaning that you can delete the message, leaving evidence of the deletion in the form of a lonely hash in the chain.

Yes, but if uncensorability is a feature you want, there's nothing stopping you (other than the enormous cost[1]) from using the messages themselves.

[1]: Not necessarily a deal-breaker. For perspective, consider that a 10-word transatlantic telegram in 1860 cost the equivalent of $2600 in 2013 USD.

Another solution is to store a URL and a integer score (so you can rank companies). The longest URL I was able to get through CloudFlare recently was about two thousand characters but most will be much smaller. If someone wants the review taken down they can talk to whomever is hosting the content. Tweets, Medium articles, Facebook posts etc could hold the review content itself. Want to take down 5 unfavorable reviews, then write 5 letters to 5 different companies.

As another user noted, you can encode the message into the chain though.