Constructing triangles (or other shapes) seems to be quite an obsolete topic, and yet, they feature in almost every high school math competition, to disappear completely in college. In recent years in the IMO, there have also been combinatorial geometric problems, like problem 2016/6. In my opinion these require deeper proofs that can be useful later as well, so why not teach combinatorial geometry in high school instead of constructions?
I am a strong advocate for teaching constructions. At least when it comes to high school geometry. My experience in teaching geometry to freshmen and sophomore high school students has shown me how much can be gained from constructions. Of course, it also depends on how you teach it. You can micromanage it and take all the fun out of it. Or you can be more liberal in your approach and let the students learn to use the tools and explore what they can create.
Geometry has the potential to open students up to a very different view of what mathematics is or can be. And it's a view that can be easily lost as they move on to other courses which tend to hammer in a more algebraic approach to mathematics. In the high school where I taught, calculus was really the only next opportunity for them to discover a novel way of thinking about mathematics, but, depending on who's teaching it, that perspective is easily lost.
The world outside of high school seems to fall into four basic camps: those who hate math, those who become mathematicians, those whose favorite high school math class was algebra, and those whose favorite high school math class was geometry. (Obviously these are not mutually exclusive camps.) There's a certain structure to algebra, a formality, a love of rules applied to manipulating symbols that seems to appeal to many. But there are others for whom the quest to reduce things to fundamental first principles has a strong appeal, the lovers of words and who enjoy weaving them together into an ordered fashion that can seemingly lead to new ideas and discoveries ("seemingly" since philosophically what's happening is hard to pin down). Then there are the students who have a more artistic bent: they like to draw, to make the abstract concrete, to view things in how they relate spacially.
Students who may have hated algebra can fall in love with geometry (I had a student once draw a picture of the week: each week she drew a new geomeTREE that somehow embodied the ideas covered.)
Constructions create a kinesthetic approach mathematics. Many young minds have a hard time understanding the diagrams in the textbook. Taking time to draw the diagrams helps them dissect what's going on and make connections. Constructions show the students how to reproduce those diagrams exactly as they appear in the textbook. The constructions help to remind the students of the various relations they should be looking for: congruent segments, congruent angles, parallel lines, bisectors,.... But also, if they are encouraged to experiment and play, they can create all sorts of art on their own. They then can discover tessellations or fractal mathematics and from their art new lessons in mathematics can be explored.
A regular emphasis on constructions creates an atmosphere in the classroom where it's a place to draw and doodle. And I would encourage that. Keeping a keen eye out for the sorts of doodles the students are doing, you can direct the students subtly in different directions. And then, if enough students doodle and the work adorns the walls, the students begin to create abstract art with mathematics embedded within. New relations arise. The students are, to a degree, the ones discovering mathematics. How easy is it to encourage such exploration and experimentation in an algebra class? (Not in my experience.)
The constructions also break students out of the bubbles of what they might consider themselves capable of. At the beginning of the year when I would throw them into constructions, students would apologize for their poor constructions by saying things like, "I'm not an artist". But by encouraging them to keep trying and not spending just two weeks out of the school year on constructions, but revisiting the topic week after week throughout the school year, by the end of the year all the students have improved greatly and a significant portion of the class indulges in doodling to a degree I don't usually see in my math classes.
In the end, you can get students for whom Vi Hart can become a hero of sorts and encourage them to see the world in an entirely new light.
Are constructions a life-long skill the students need? Probably not. But by introducing the students to this approach to geometry, vista open up for them particularly in a phase of their maturity when abstract reasoning is unfamiliar territory. The connection between abstraction and understanding is made manifestly concrete in their very hands.
I've recently committed, in my college algebra classes, to presenting proofs for as many or all topics that I possibly can. This has made me much more aware of how often basic geometry constructions (transversals, etc.) are required. This includes: Pythagorean theorem, distance formula, midpoint formula, that any line is a linear equation, slope is constant on a line, parallel relationship, perpendicular relationship, circle formula, etc.
In conjunction with that realization, and in trying to introduce students to the idea of general proofs in college mathematics, I find that I'm building more explicit bridges between college algebra and the high school geometry course -- reminding students of that course, building upon it, and highlighting what should have been the primary message (importance and accessibility of proofs).
In short: Yes, the basic geometry constructions and proofs are fundamental to later work in more than one sense (both the results themselves, as well as the introductory idea of mathematical proof).
I suggest that emphasizing plane geometry problems and their solutions could be more effective than a systematic introduction to constructive geometry. For example:
Alexander Shen. Geometry in Problems. MSRI Mathematical Circles Library. Volume: 18; 2016. AMS link.
Here us a typical problem (out of $750$ problems):
I strongly believe in teaching construction problems. They can be used to stimulate lively discussions in the class. For example, see the several construction problems in Geometric Transformations, Vol I by Yaglom (MAA publication) (http://maa.org/press/ebooks/geometric-transformations-i). Also, given orthocenter, circumcenter and one of the vertices, is it possible to construct the triangle? Is there one or many solutions? These kind of questions help students understand Euclidean geometry much better and at the same time teach "problem solving".
I'd like to note that construction geometry can be very helpful for IMO level geometry problems.
I always make a large diagram (picture) and construct all lines and points given in the problem as precise as I can, so that I can develop conjectures on what points be on one line, what quadrilaterals might be cyclic, which two lines might be parallel, etcetera.
Doing this precisely is only possible if you know the constructions well. Also, I like how constructions apply knowledge of geometry, for example, when proving that the construction is correct or when even thinking of efficient constructions yourself.
It is true that combinatorial geometry is gaining a lot of popularity. Note that IMO 2015/1 was also combinatorial geometry. On the other hand, I think that combinatorial geometry is more combinatorics than geometry, so I would be careful when replacing geometry with combinatorial geometry since you are then actually replacing geometry with combinatorics.