You wrote:
"Take modern genome sequencing. Practically every way that the genome data *could* have come out would have falsified common-descent. Case closed. Done and dusted."
Are you saying that science has proved that God did not create man in His image, or if He did science has proved that He used evolution to do so and that it took circa 3.5 billion years?
Not having falsified common descent doesn't mean you have falsified common creation, does it?
And yes, there is disagreement (NOT "absolutely unambiguous statements") over whether there is a LUCA or not among those "in the know" AKA those who make educated guesses.
Last universal ancestor (LUA), is the hypothetical latest living organism from which all currently living organisms descend. As such, it is the most recent common ancestor of the set of all currently living organisms. Also LCA (last common ancestor) or LUCA (last universal common ancestor). It is estimated to have lived some 3.5 billion years ago.
Science cannot disprove God's actions, since God can do anything, but yes, if God created man He (a) used evolution to do so, or (b) He planted ubiquitous false evidence that He used evolution to do so, or (c) He permitted an adversary to plant false evidence. Since Christians prefer not to call God a liar the huge number of Christians who accept evolution believe (a).
And yes, there is disagreement (NOT "absolutely unambiguous statements") over whether there is a LUCA or not among those "in the know" AKA those who make educated guesses.
The nature of that disagreement is related to information at the boundary of our knowledge and subtle distinctions about the exact nature of early life on earth. That is an answer that we may never know, given the remoteness in time and the paucity of physical evidence of exactly what happened. It isn't related to whether or not the modern biological kingdoms share common ancestory, since assuredly they do. The fact that (at the very least) the overwhelming majority of life currently existing on our planet shares common ancestory is not currently scientifically disputed.