This would make more sense as a narrative if there were any prominent examples of short sellers materially harming the operations of a company, but I can't find any. In theory short sellers could help push a company into insolvency, but so could a lot of other vicious cycles for a failing company- worse terms from suppliers, worse credit terms, less trust from customers.

The Wikipedia page on "bear raid" mentions a single example, from 1609. https://en.wikipedia.org/wiki/Bear_raid