I think the misunderstanding here is that `X` is different from `X is known,` and `X` doesn't imply `X is known.` Consider if we are analyzing a program which does not halt (`~doesHalt`). The halting problem tells us that we can't actually prove this (`doesHalt is not known`).

Can we flip the question around? Under what scenario would you accept the middle condition of, "we do not know whether X is or is not possible" to exist?

There are plenty of things you can say "we don't know whether or not this exists." But that doesn't somehow invalidate LEM. Whether or not something is possible is a binary state, regardless of whether or not we know the answer.

For instance, we don't know whether God exists or not (just a contrived example). Now consider the statement: "It's not impossible for God to exist." That's equivalent to saying "It's possible for God to exist." The two statements are totally equivalent. Sure, we don't know whether or not God exists, but that doesn't make the statement "It's not impossible for God to exist" fundamentally different from "It's possible for God to exist."

Now consider the headline: "I think faster than light travel is possible." And the suggestion is to change this to "I think faster than light travel is not impossible." Again, the two statements are totally equivalent because whether or not something is possible is a binary state and the law of the excluded middle.