Essentially, Yelp provides a lot of utility to the users, but can't make money off them. It provides leads for the shops, but it doesn't control the demand side. Its review data is scraped by Google, so the value of its data is diluted. I think Yelp is a failed business model by itself. The only reason I am not shorting it right now is I think someone will acquire the company soon. In the hands of a big company, it could act as a huge loss leader for other value added services.
It isn't clear to me that Yelp, as it exists today, can ever be profitable -- they need to hire expensive sales people to cold-call businesses to sell ads that run at CPM rates that are already incredibly high. Yet they can barely break even. It's not going to be pretty as their ad rates are forced to become market competitive.
There may not be any traffic after they slap a giant Samsung ad on their site.
I've also noticed cases where legit looking reviewers have complained about their positive reviews being removed.
Yelp doesn't change search rankings based on advertising. Source: I worked there, on search. It isn't true.
Looking at the search results for pizza sorted by the "best match" algorithm is mind boggling.
No, it doesn't. Setting aside the fact that you haven't actually done the analysis (so you couldn't find a correlation if you wanted to), it's an example of "people who want to believe in a conspiracy will find evidence for the conspiracy", with a nice helping of the post-hoc fallacy: Yelp has a huge army of sales people calling businesses every day. Rankings are also updated 24/7. It is inevitable that someone will be called, at random, right before they experience a ranking change.
A search as generic as "BBQ" or "pizza" is going to have a lot of variation, because people are constantly using those tokens in reviews, and they match across a huge number of businesses. I very sincerely doubt that there are 6-page differences in a business' ranking on a day-to-day basis, but if it does occur, it has nothing to do with their advertising status.