He said he had to fight management to get this changed, as they didn't want to spend the money to modify the hard-coded applications to use a more secure password or to use Windows authentication. Unbelievable.

There will often be someone held accountable, even if the issue isn't publicized, and that person could easily be you. Follow best practices for security, raise awareness where you can, and document the exceptions with your objections noted. It might not make the company more secure, but it likely will help you keep your job.