Perhaps because it gives established players a big leg up in terms of compliance costs (imagine being a small news site having to create this searchable system etc)
Your right to anonymous free speech does not mean Facebook is obligated to let you express that right, only that the government can't restrict your right to anonymous free speech. No reason you can't set up your own anonymous website.
At this point, it's worth it.
If laws protect people. more laws = better? What is the downside of passing any law?
In this case I could face discrimination at work if they found out that I was buying ads for my anti abortion website.
No this is a question about a specific law. Making it a general point about all laws is disingenuous. Why do anything ever?