Short answer: The skewed content is not a good reason for avoiding IMO-style contest training, because if the training is done right then the students will be led to explore mathematics and would never have a mistaken picture that mathematics is mostly about IMO topics. (I of course compare between good IMO-style training and good teaching of university-level mathematics, because there is no point comparing lousy training with anything.)
Long answer: (Note that the question has been edited slightly so my responses were to the original points.)
I believe there are concrete mathematical pedagogical issues relevant to your question. A major point is that IMO-style mathematics is a significant step up from the blindly rote-learnt mechanical 'math' that pervades many high-school education systems (both in the syllabus and in the delivery). This is because there is a proper focus on rigorous logical reasoning in the form of proofs. I think that this is the most important reason to value IMO-style problems, and should be properly emphasized in any competent training on IMO-style problems (whether or not with the goal of participating in contests):
Students should be taught logical reasoning in conjunction with how it is used to solve IMO-style problems (starting with easy problems but with fully rigorous solutions). The objective is that they should be able to carefully, accurately and confidently identify what exactly is proven, what they wish to prove, and what assumptions they used, throughout the solving process, and similarly perform the same analysis at any point in someone else's proof.
Students should be trained to write logically valid proofs, not just to write handwaving arguments or a messy bunch of statements or equations with imprecise logical structure. If any deductive steps are skipped in a proof, the student must be able to produce upon being asked a correct subproof to justify the jump.
Students should also be trained to analyze others' proof attempts and correctly verify them or identify logical errors or gaps in the argument. Even if the teacher claims that a proof is correct, but actually is wrong, the students should be able to confidently reject the teacher's claim. In other words, an incorrect argument from authority must lose to the students. This is a litmus test that the students actually know what they are doing.
I also want to address several misconceptions revealed by some points in your question about competitive mathematics. You are definitely right on certain points, such as the fact that contest mathematics tend to be restricted to a small 'syllabus'. This is both a disadvantage (as you noted) and an advantage, because contest problems are usually much more interesting than 'standard' mathematics. But in my experience not all your points hold.
Nobody disagrees that 16th-century tricks of Euclidean geometry are not of much use to any professional mathematicians today. Some even doubt if it should be part of the high-school curriculum at all.
Well, here are some that I think professional mathematicians ought to know.
The compass-and-straightedge construction of the square-root of a given ratio of lengths is directly relevant to the proof that the constructible reals are precisely those that can be obtained by a sequence of quadratic extensions. This result is an important one in the history of Galois theory, and I think it should be taught in any introductory course to field theory.
Another one is the proof of the Pythagoras theorem, which is needed to give a real motivation for the modern notion of Euclidean spaces. Related are all the applications of circle and conic geometry in classical mechanics and optics, which is very much a part of applied mathematics.
Yet another often overlooked tool in geometry is inversion, which is closely related to the Mobius transformations in complex analysis, as well as the more general conformal mappings.
In general, competition stuff are irrelevant to the research of most of the professional mathematicians (and of course, more irrelevant for other people).
Let's look at the IMO syllabus specifically, which roughly covers elementary number theory, combinatorics, graph theory, inequalities, Euclidean geometry and functional equations. In my own experience, all the concepts in number theory, combinatorics and graph theory in the IMO syllabus also show up in any university-level courses in those topics, and these form foundations for higher-level courses. It is true that many things in the other topics do not show up in university courses, but still many do.
For example, the AM-GM inequality is a basic one that every mathematician must know, and higher mathematics also relies on some IMO-level inequalities such as Jensen's, Bernoulli's, Cauchy-Schwarz, Chebychev, Holder's, and the power mean inequality. The concept of smoothing is also a very important and productive one, as are the various other generic techniques commonly used in IMO including homogenization, re-parametrization, splitting of the domain, even if we scrupulously avoid whacking techniques that many IMO students learn such as Lagrange multipliers.
I have given some examples of Euclidean geometry applications above, but one major advantage of learning Euclidean geometry is that it is a beautiful and vast playground suitable for learning rigorous logical reasoning with focus on propositional logic, as almost all Euclidean geometry problems can be solved in pure propositional logic.
As for functional equations, I would say that some of it (such as learning the solution of well-known functional equation $f(x+y) = f(x)+f(y)$ for continuous $f : \mathbb{R} → \mathbb{R}$) provides the IMO student a glimpse of real analysis. The concept of transforming the function itself is also a very important one (such as in reducing the functional equation $f(x·y) = f(x)·f(y)$ for continuous $f : \mathbb{R}^+ → \mathbb{R}$ to the previous one), and such techniques show up in all branches of mathematics.
Although students are likely to learn some knowledge beyond high-school if they manage to get to IMO level, in general, for most students who are training for competitions, the contests tend to limit them within the scope of high-school conte[s]ts. This, again, creates a biased picture of the subject in their mind.
You may be right that training for only high-school contests, if they do not involve proof questions, would create an inaccurate picture of mathematics. However, in many countries there are high-school contests with final stages involving proof questions. It would be sad if students merely trained to get some results on only multiple-choice or short-answer questions. As for IMO level, most IMO participants I ever got to know did not ever stay within the 'confines' of IMO-style mathematics, but on their own delved deep into other topics in mathematics as well.
Some people do not like the fact that maths is made into a "sport" - the long hours of training are laborious, but does not necessarily lead to deep thoughts.
I do not agree with excessive training in only one thing, whether it is mathematics or not, if other important subjects are neglected. On the other hand, long hours of training is typically necessary to become an expert at anything. Some people say that an average person needs roughly 1000 hours to become reasonably good at something, and 10000 hours to become an expert, and I think it is roughly right.
students who are good at maths would better not pursue their interests through competitions; instead, they can learn higher level university-level maths in advance.
I have some gripes with contest mathematics (such as the fact that many of them including the IMO use problems that have appeared before), but I would say that there is much benefit in learning and even training for IMO-style mathematics. As I mentioned earlier, contest mathematics is usually far more interesting than modern mathematics to a high-school student, so there is ample motivation to explore.
Also, in my opinion, it is a rather 'clean' playground. Earlier I mentioned Euclidean geometry as being suitable for learning propositional logic. Even the other IMO topics are more suitable than modern mathematics for learning first-order reasoning.
For example, it is easy to learn induction, strong induction, structural induction, well-ordering, and the extremal principle in the context of combinatorics and graph theory, and there are numerous cute problems involving these tools in their solutions. I personally consider full first-order induction as a key part of logical reasoning, but sadly numerous undergraduate math majors cannot use it properly.
