Ok, I now see where your doubts are. I think it can be shown why the territory unit is the correct one.
Consider the 9x9 opening analysis I linked earlier. The territory one shows what are the correct W opening moves, and what are suboptimal moves that lose 1 point. The area analysis shows that all those point-losing W openings are still “perfect” under area scoring (and @Samraku is pleased by thinking there are more viable openings under area).
But the problem is that the rounding (OR 0x01) happens only on the final score. So it is not individual mistakes that will get masked, but their cumulative sum. For imperfect players, losing a point of territory this early WILL also reduce their expected area score EXACTLY by 1 point (measurable by playing lots of area games from the position for example) - contrary to what the area analysis shows.
Also think about this duality view. The game IS always about surrounding more territory, regardless of rules. And even area scoring “sees” the difference between B+7 with an extra stone or without it, it just cannot express this difference in its numerical result (w/o button).