Okay, I seem to be ranting too much in comments, so let me try to put forth my points and opinion here (as a community wiki, since it is opinion).
First, to address the question: What is the significance of straightedge and compass constructions within mathematics? Historically, they played an important role and led to a number of interesting material (the three famous impossibilities lead very naturally to transcendental numbers, theories of equations, and the like). They are related to fascinating stuff (numbers constructible by origami, etc). But I would say that their significance parallels a bit the significance of Cayley's Theorem in Group Theory: although important historically, and relevant to understand the development of many areas of mathematics, they are not that particularly important today. As you can see from the responses, many find them "fascinating", many find them "boring", but nobody seems to have come forth with an important application.
Now, addressing the issue of teaching it at K-12. Let me preface this by saying that I am a "survivor" of the New Math, which came to Mexico (where I grew up) in the 70s. I would say I became a mathematician in large part despite having been taught with New Math, rather than because of it. Also, I did not attend K-12 education or undergraduate in the U.S.; I am a bit more familiar with undergraduate at the level of precalculus and above thanks to my job, but not very much with the details of curricula in sundry states in the U.S. So I may very well have a wrong impression of details in what follows.
Now, one problem, in my view, is that when we talk about "math education", we are really talking about two different things: numeracy and mathematics. This is the same phenomenon we see when we think about "English class". I suspect that English Ph.D.s are nonplussed at people who think they spend their time dealing with grammar, spelling, punctuation, etc, just as mathematicians are nonplussed that people think we spend our time multiplying really big numbers by hand. English education has two distinct components, which we might call "Literacy" and "Literature". Literacy is the component where we try to teach students to read and write effectively, spelling, punctuation, etc. Making themselves understood in written form, and understanding the written word. On the other hand, Literature is the component where they are introduced to Shakespeare, novelists, book reading, short stories, poetry, creative writing, historical and world literature, etc. We consider English education at our schools a failure when it fails in its mission with regards to literacy: we don't consider it a failure if students come out not being particularly enthused with reading classic novels or don't become professional poets, or even if they don't care or like poetry (or Shakespeare). The professional English Ph.D. engages in literature, not in literacy. Literacy is the domain of the grammarian.
Mathematics education likewise has two components; numeracy (to borrow the term from John Allen Paulos) and mathematics. Numeracy is the parallel of literacy: we want children to be able to handle and understand numbers and basic algebra, percentages, etc., because they are necessary to function in the world. Mathematics is the stuff that mathematicians do, and which we all find so interesting and beautiful.
"Mathematics" also includes advanced topics that are necessary for someone who is going to go on to study areas that require mathematics: so the physicists and engineers need to know trigonometry; business and economists need to know advanced statistics; computer science needs to know discrete mathematics; etc. Much like someone going on to college likely needs more than simple command of grammar and spelling.
Part of the problem with math education is that it so often conflates numeracy with mathematics; another part of the problem is that mathematics was included in the "classical" curriculum for much the same reason as Latin and Greek were included: historical reasons, because every "gentleman" was expected to know some Latin, some Greek, and some mathematics (and by "mathematics", people meant Euclid). To some extent, we still teach geometry in K-12 because we've always taught geometry. But it is not part of the numeracy curriculum.
It would be wonderful if we had the time in K-12 to teach students both numeracy and mathematics; it is a fact that today we are failing at both. We don't perform better in teaching students numeracy by teaching them beautiful mathematics, even if they understand and appreciate them, just like making them really understand and care for Shakespeare's plays (through performance, say) will make them able to understand a set of written instructions, or write a coherent argument.
Trying to excite students about mathematics is all well and good; but numeracy should come first. Trying to excite students about the wonderful world of books, plays, and poetry is all well and good, but we need them to be able to read and write first, because that's part of what they will need to function in society.
Constructions by straightedge and compass can become an interesting part of mathematics education, just like geometry. I just don't think they have a place in numeracy education. But they are taking the time we need for that numeracy education. Trigonometry used to be part of the numeracy education that people needed; this is no longer the case today. Trigonometry, today, is a foundational science for more advanced studies, not part of numeracy. But we are still teaching trigonometry as a numeracy subject (hence the rules, recipes, mnemonics, and the like). We could try to turn trigonometry into a mathematical subject, sure; or we could postpone it until later and only teach it to those for whom it is an important foundation.
Added. To clarify: I don't mean to say that the "solution" is to do less math and more numeracy. I think the solution is likely to be complicated, but the first step is to identify exactly what parts of what is currently branded as "mathematics" are really numeracy, and which parts are mathematics. Trigonometry is taught as part of "advanced mathematics", but it is taught as rote and rules because it was really numeracy. We don't need to teach trigonometry as numeracy any more, so we shouldn't. If it is to be taught, it needs to be taught in the right context. Geometry is similar: geometry used to be taught as basic numeracy because "every educated person should know geometry"; (of course, "educated" at the time meant "rich and land owner, or with aspirations in that direction"). We don't need most of geometry as basic numeracy, we want it now as mathematics. So teaching geometry as numeracy is a waste of time, and it takes up the numeracy time that should be spent in other things. (It can still be taught during the mathematics time). I certainly don't say "drop all the math, concentrate on the numeracy". I say, "when dealing with numeracy, concentrate on the numeracy, not the math, and don't confuse the two." Nobody seems to confuse spelling rules with reading novels, because we separate literacy from literature. Too many people confuse arithmetic with mathematics, because we don't separate them.
The reason I talk about dropping trigonometry and doing some basic statistics is precisely that: trigonometry is being taught as part of the numeracy curriculum, when it shouldn't. Basic statistics, say at the level of the wonderful How to Lie With Statistics, is not taught as part of numeracy. But in today's world, there is a far better case for statistics being part of the basic numeracy education than trigonometry. Everyone coming out of High School should know that taking a 10% pay-cut and then getting a 5% raise does not mean you are now at 95% of your old salary (go do a spot check, see how many people think you are). They need to know the difference between average and median, so they are not misled by statements about "the average salary of the American worker". They should understand what "false positive" and "false negative" means. They should be able to interpret graphs (even the silly ones on the cover of every USA Today issue) and be able to spot the distortions created by chopping axes, etc. These are numeracy issues.
Likewise high school geometry: it is trying to be both numeracy and mathematics, and I think it generally fails at both. There are some components of geometry that are part of numeracy, they ought to be treated that way, but the parts that are mathematics should be separate.
One problem I have with Lockhart's lament is that he does not make clear the distinction between numeracy and mathematical education. The nightmare he paints for the musician and the artist is precisely that musical education is being turned into the equivalent of numeracy/literacy education, thus doing a disservice to music-as-an-art.
The science of mathematics that we all know and love has some intersection with numeracy and arithmetic, but we all know it is a limited intersection; just as the study of literature has some intersection with the study of grammar an spelling, but the intersection is limited. The main purpose of K-12 education (or at the very least, K-6 or K-8) should be numeracy, with some limited forays into mathematics (just as the main purpose will be literacy with some limited forays into literature). The way to teach numeracy is necessarily different from the way to teach mathematics. Numeracy requires that we memorize multiplication tables, for all the horror this will cause to modern education people; this of course is a far cry from teaching the mathematics of multiplication, which may very well be very interesting and awaken the child's curiosity and wonder at the world. That can be done within the context of mathematical education, but it shouldn't be done in the context, and at the expense of, numeracy.
Trigonometry used to be part of numeracy; it no longer is. It is now either mathematical or foundational for advanced studies, so it should be treated as such. Geometry used to be taught for reasons which no longer hold, and to some extent we continue teaching it as a historical legacy; we shouldn't. Those components which are numeracy should be taught as that, and we could move the rest (including constructions with compass and straightedge) to the more creative, mathematical education side of the equation.
Anyway, I've ranted long enough, and probably made myself a few detractors along the way...