My first inclination is to read letters like this as a threat from employees to the employer. It says hey boss-men, this shite is not on. Signing anonymously undermines that message. I tend to read those signatures as as, I don't like this but it's not worth my job. I have no faith in the efficacy or even existence of "obstruct or delay" tactics from folks like that.
No it doesn't. It says "Hey boos I'm telling you this shit is not cool, and there's nothing you can do to me personally because you don't know who I am."
Let me put it differently. Suppose YOU are the boss. You company has 1000 employees and you receive a letter with 500 anonymous signatures saying "we fucking hate what you're doing" (so, 50% of your employees, 100% anonymous). Do you get a little bit worried? Or do you get not worried at all because everybody signed anonymous? Actual question, let me know how you think.
Why does this change the calculus for management? They don't pay folks to be happy, they pay them to do their jobs. Threaten to take away the labour however and you create a bargaining position. That's how strikes and threats of strikes work. This letter is fundamentally different. For a start, have you considered the veracity of a list of anonymous petitioners? How do you differentiate the real thing from a made up list?