Analogies are always risky business ;)

Facebook has a public service and one of the options is to Unfollow; someone wrote a browser extension to do this automatically for all items.

Fundamentally, what is the difference between automating this process and doing it manually?

In your example of a fax machine, arguably fax numbers are a private entity; there is no requirement for publishing fax numbers nor is fax automatically publicly listed for everyone to see. A malicious spammer would need to either obtain the fax number from a listing somewhere or brute-force the number, and similarly, the only way to __know__ that a fax has gone is ambiguous. obtain the fax number from a listing somewhere or brute-force the number, and neither is really analogous to what a browser offers.

I think your analogy conflates a few concepts incorrectly, namely that there is some unexpected or undue financial consequence to Facebook for publicly allowing users to Unfollow Groups; if the extension __needlessly__ generated traffic, this is closer to your analogy. But as I can see how the extension works (based on archived copies found on shady sites), it's not undue traffic, it's just expediting the process of manually Unfollowing groups.

Facebook shouldn't have a recourse here as I see it; the automation causes no undue burden on facebook that isn't possible by manually clicking, an arbitrary review of the extension suggests there is no undue stress on the servers that differs in any way from the traffic one might generate if they manually unfollowed groups. Automating the process indeed might be undesirable for Facebook in some way, but fundamentally the same result is achievable with manually clicking, and I think a more substantial evidence of damage is required from Facebook to justify such a threat.

If we take it to a logical comparison, should Facebook have the right to block a mouse + keyboard automation tool that I script to react at human speeds but is pixel-perfect to unfollow groups?

If the answer to this from Facebook is "yes", then the natural question is "what is the similarity between these processes?"; if the answer is "automation", then the natural question is "why is this damaging to Facebook as opposed to me just manually unfollowing??", and I'm not confident Facebook has a reasonable/strong answer to this.

If Facebook is fine with the slower method, then the question becomes "what is the real concern with the faster method? I will skip the logical follow-ups here as the response is already long.

Facebook should __not__ have the right to sue just because they don't like an activity; no one benefits from this; quite the opposite, smaller parties are actively harmed by such behavior as they lack the financial resources or confidence (or both) to respond to such a legal challenge, and this was never the intent of law. One should not need heavy financing to secure their natural rights; if Facebook wants to position that the extension is somehow illegal as per terms of service, I think the duty is on them to demonstrate how it's significantly damaging and how it differs from a dedicated person armed with a cup of coffee and an hour of free time; if Facebook cannot make a significant distinction outside of convenience for the person, then I don't see a basis for legal recourse.