I know it's fun to hate on the big tech companies recently and act like they are bullies (and indeed in many ways they are), but this is a bad example of that. Most companies I know of would outright fire you if you, against explicit company policy, went to the press and started badmouthing your employer. The fact that Amazon only gave her a warning is the only surprising thing in this article.
First off we are talking about benign matters like employees' opinions, not trade secrets (common sense would prevent the latter from leaking).
Second, I am not advocating to force anyone to disclose anything, but merely for them to have the potential to do so if they wish. If you trust them enough to complete the task then surely you should also trust their common sense about what to talk and not talk about (trade secrets, etc) to the press?
If the thought of anyone being able to "ruin" the company's reputation by airing dirty laundry is scary, then maybe fix the root cause instead of the symptoms and ask yourself 1) why is there dirty laundry to begin with? and 2) why would the employee be unhappy enough to leak it?
There are tons of happy small businesses with no such thing as "PR" nor media policy (just employees' common sense) and they don't seem to implode, because either there's no dirty laundry or the employees are paid & treated well enough to not put the company in disrepute. It's like mutual trust & respect.