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.
Yelp doesn't change search rankings based on advertising. Source: I worked there, on search. It isn't true.