I’ll bite, since no one else seems to have - 1492, Columbus discovers North America and 1776, War of Independence
I’d say it has another, very important property: Preparing the student for working in fields of math which are more abstract, such that you can’t appeal to “common sense” or “real life” even in principle, and have only the given information and the axioms to work with.
And, as I said, this is a fundamental, terminal problem for some people, a pons asinorum in close to the original sense: The phrase pons asinorum means “ass’s bridge”, with ass as in the four-legged donkey-type animal, and refers to the concept of a bridge which the ass will not (or cannot) cross, such that a pathway for some is a barrier to them. (This originally applied to a specific theorem, that the angles opposite to the equal sides of an isosceles triangle are themselves equal, but it’s more useful in the expanded sense.) In this case, the bridge is between having a specific physical model in hand and moving on to a more abstract, axiomatic understanding of mathematics. Weakening the diagrams, making them more abstract as opposed to accurately representational, can be a very gentle way to do that.
Is it the actual goal here? Eh, it’s more likely they didn’t take the time to make things perfect because there’s no margin in it. This is still an important point.
I used to remind my Geometry students that “diagrams lie.” The point I was making was that they couldn’t just rely on what the diagram seemed to be showing for the purpose of determining an answer. In the US, Geometry is not just a course in spatial mathematics, it’s also a course in deductive reasoning. So everything that you do with the course, including problem solving, is based upon a series of assumptions (postulates) and deduced conclusions (theorems). So the problem in question becomes solvable without reference to whether or not the diagram is scaled properly.
Of course, as a teacher making up my own diagrams, I always tried to make them as accurate as possible, because a LOT of students cannot shake treating the diagram as containing information based upon how it looks.
As you imply, in the diagram, BAC appears isoceles, although there’s nothing to indicate that that is the case. As you show, this allows the student who’s forgotten the central angle theorem to prove a special case of it.
I’d have drawn the problem diagram so that BAC appeared clearly Non-isoceles. But this would have a drawback: the student could safely infer the full central angle theorem he’s forgotten — because without the truth of that theorem there’s not enough information to find the angle BAC.
To get around that flaw, I’d have shown the angle ABO as being 8° or such. Total red herring since the angle BAC is independent of ABO. Am I a sadist?
On the subject of cheating shortcuts… When I first read the question, I didn’t pay attention to the quadratic equation. I was convinced that the poster was thinking of a pair of related events. My instinctive guess was 1914 & 1939, the years in which the World Wars started, which was way off, of course. 
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Heh heh … I thought the same thing but without thinking of the possibility that the events were related. I just thought “Two dates of events in Western history … what are the first two important dates in Western history that come to mind? Probably 1776, that’s an important one. And, um … oh, and 1492, when Columbus sailed the ocean blue?” A glance at the second coefficient in the equation, which was 3268, seemed in the right ballpark to be 1776+1492, and that seemed like enough effort to expend.
I like the problem actually. It’s trivial for 9th grade geometry.
And multiple approaches work to solve it.
I will go back to this as a Geometry teacher. The reason such artificiality of algebra is inserted is solely to make certain that the students don’t forget their Algebra I equation-solving techniques while taking Geometry. I, too, find these problems annoying, mostly because if they are given, then they have to be graded on the basis of correct Algebra solutions, which, of course, I’m not teaching in a Geometry class. But my fellow Algebra II teachers get quite perturbed if my Geometry students take their class after a year of mostly algebra-free exploration of Geometry.
Now, admittedly, one can construct problems that are more integrated in weaving algebra into the geometry, but that takes time, and often requires that the problem be a word problem of some sort. Problems like the OP posted are exercises that will come 30 - 50 at a time at the end of each section of a chapter, to drill the concepts involved in the unit. Not shockingly, they tend to be devoid of meaning. 
I still remember my very first ever college physics test. My mind had completely blanked: I couldn’t remember anything at all but F = ma. And so I re-derived everything from that. Only to finally turn over the last page, and you can imagine what it was. Although, let me tell you, once you’ve done that once, you’ll never forget those formulas again.
(as an aside: I thought you were a lawyer. How did an Esq come to be a teacher?)
Back in the mid-to-late '90s, I bought a soul (Walmart, 50% off), and that eventually forced me from the legal profession. 
After some flopping around looking for something else more worthwhile to do, I spent a couple years being a substitute teacher (all grade levels). I decided I liked the profession enough to become a trained teacher.
While social studies might have made more sense (given my original undergraduate B.A. in PoliSci, every coach at the high school level teaches social studies, limiting job availability. I decided that mathematics was not only an area with great demand, but an area that cried out for teachers who actually knew how to express themselves in the English language, since too many HS math teachers end up explaining a concept the same way over and over, then tagging those who don’t understand “bad” at math. Since I had been a math-inclined student in high school (Mathletes, etc.), I figured it was a natural for me.
This last fall, after some 10 years in the profession, I decided to retire, and live the good life of being fancy free.