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.
Or how about:
Since God created the universe, and since He made statements about the origins of mankind and the history of mankind, and since we believe God is moral and truthful and would not lie to us or deliberately contradict His own statements about Himself or the universe He created:
A. We believe His statements to be true and strive to the best of our ability to understand and obey them.
B. We do not believe His statements to be true, but still want to believe in Him.
C. We reject the statements of God in the Bible as truth.
If A, then perhaps we have misinterpreted the data based on assumptions about what happened in the past (uniformity of natural causes in a closed system). We cannot re-create these conditions in a lab experiment, so we have to rely on our assumptions, or admit that perhaps conditions were not always constant, or our "baseline" may be incorrect. If the data contradicts clear Biblical statements, we have to abandon the Bible or "science", or "do" a systhesis that leaves the Bible neither the Bible nor science science. If A, then we must keep science inferior to Biblical revelation when they contradict, since scientific observations and interpretations are incomplete and subject to human fallibility.
If B, you have much of modern theology, where we have created God in our own image and can pick and choose from the Bible the statements we wish to believe as it suits us.
If C, "let us eat, drink, and be merry, for tomorrow we die."