For most companies that is true. It is not true for companies that need to access the capital market frequently, like most financials or even MLP that have financing needs to that access capital markets for debt and equity.
Financials in particular need to roll over debt, over gibt credit etc literally every day and and their reputation is extremely important in the view of their stakeholders (counterparts, customers, depositions, creditor). A severely lower share price and undermine confidence na impact the business very quickly. That’s the reflexivity principle. For other companies (industrials, even tech) that is much less the case.
My understanding is that the scenario you mentioned is the problem with the lower stock price - which maps to the low equity value for the company. But does shorting bring down the price of the stock artificially? (The news about someone's short position or badmouthing about the company could bring down the company. But does the process of shorting - will it bring down the stocks?)