I'd be surprised if a random member of the public could even define what free software is. They'd probably think it's connected to the cost of the software rather than its freedom giving properties.
That said, I think that the view that with enough eyes all bugs are shallow is false. Given that bash is used in millions and millions of servers and the bug took decades to root out, we must think of a better way to get eyes on the code.
The whole stack needs a line by line review by security experts. That will cost tens if not hundreds of millions of dollars but my view is that it's probably worth it. Then we have to make sure all changes get reviewed in the same way.
The result of this process would be a super-hardened version of OpenBSD. It would come with a nice fat government certification and if you want to do business with the government, you have to use that distro.
That might rub people up the wrong way but I think that's what's ultimately going to happen eventually. A lot of this infrastructure is so critical to the modern economy that we can't just run any old code anymore.