The "American Rule" also leads to many lawsuits of questionable merit being brought against corporations (and wealthy individuals). Attorneys take these cases on contingency, knowing that the defendant will likely just settle rather than incur a lot of legal fees, and the firm will take 33-40% of the settlement. These "nuisance lawsuits" are a big problem for some industries. If there was universal fee-shifting, these suits would be discouraged.
It can be worse though. In some jurisdictions and for certain types of claims (e.g., employer-employee wage/hour suits in California), statutes provide for "one-way fee shifting". One-way fee-shifting means that if the plaintiff wins, the defendant pays the plaintiff's legal fees, but if the plaintiff loses, the plaintiff does not pay the defendant's legal fees. Predictably, this leads to some law firms specializing in these types of cases, taking cases on contingency, discouraging the plaintiff from settling, and racking up huge legal fees that will be paid by the defendant if they win. And the law supports this. I read one article about the California Court of Appeals upholding a $250,000 fee award in a case over $20,000 in unpaid wages.