No because we'd actually act upon it because of purely logical chains (1) of cause and effect, not free will.
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(1) Or rather "networks" (of cause end effect) that work through all of the (currently 4) dimensions.
First of all, the discovery itself wouldn't be an action of "free will".
Second, who said we can act on "purely logical chains of cause and effect" when there's no free will? Whether we act on those chains or not will already be predetermined by the "no free will" mechanism determining our actions.
If anything, as societies we act pretty illogical a lot of the time (heck, even most), for stuff scientifically known to be BS.
> as societies we act pretty illogical
It just looks that way on the surface. Below the surface, you can track all decisions back to their causes (down to "the smallest" scope).
Not really, it doesn't matter. If there's no "free will", then we have no say on whether we judge people or not, not even as to whether we believe in free will or no.
Your answer works accidentally on two conflicting levels, assuming that if we discovered that we have no free will we'd still have free will to act on a certain way upon it (not to mention that the discovery itself wouldn't be on our own free will).