If your business includes software in any meaningful function, you are making money from it. Any competent lawyer would be able to successfully argue that. Charging money for a license is not the only way, otherwise everybody would just switch to charging for "consultancy service", which coincidentally provides free software license, and avoid any liability.
> You would be free to publish software, open source or not, without incurring liability as long as you don't make money from it.
You as a private person would be. That's my point - that would be the only way to do open source, any corporate support of open source projects would imply full liability, which would be impossible for a product the company gets to revenue from. It would be much harder for a business to justify supporting an open source project when liability costs are added to the equation.
> The currently widespread practice of trusting sensitive user data to open source code without an audit (either internally or via a third-party, e.g. Red Hat) is horrifying
Audits cost money. Tons of money. And they don't guarantee anything - bugs in OpenSSL have not been discovered for years despite thousands of people using the code, poring over it and billions depending on it. There's no magic in "audit" that allows code to be bug-free after it - if there was such a magical procedure people would already be using it, but there's no indication anybody has invented "audit" procedure that allows to eliminate all bugs. Existing flawed procedures are already being used - every company that produces software that I ever heard about uses them - and they are not enough. So what would happen is drastically raising the costs (to the point where having a website would no longer be affordable to an average person) while not significantly improving security.