That is a misunderstanding of the refining process of science. Details change at the edge of our knowledge, but well-established central theories (the example under discussion being evolution) are supported by untold millions of data points, 150 years of observations and attempted falsifications. Relativity replaced Newtonian mechanics, yet for 99.99% of practical purposes Newtonian Mechanics remains true; how could it be otherwise? Newton's work survived hundreds of years of practical application and observations without anyone noticing that it was incomplete.
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. Evolution would have been dead and buried, indeed creationists predicted before the molecular data came in that it would falsify common descent. Yet what actually happened was that the genome data for every species that we sequence confirms the very precise predictions of common-descent, in spades. That is why even prominent IDers like Behe and Denton endorse common descent of life on earth; they know that the data in that area is unambiguous.
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.