Yes. We should teach trigonometric substitution. But, I take it a step further, I think we should also teach hyperbolic substitution. With this additional technique the idea of the substitution is much clearer. Also, teach it with confidence. Teach it as if they can all understand it... because they can.
Why do students have trouble with this topic? I submit, the reason is often ignorance of trigonometry. Especially an inability to manipulate basic algebraic identities. Should we face this inequity of their past education in second semester calculus? I think yes. I could throw up my hands in despair and just say they're too far gone, but, I refuse. It's not too late to learn elementary algebra and trigonometry. That is all trig-substitution entails.
Almost any topic we cover in calculus, or algebra for that matter, can be done with a CAS. However, to think that the point of calculus is to solve calculus problems misses the mark. The real reason is that these students (I am not talking about the life-science calculus crowd here, rather those students destined for science-heavy technical majors or math) need to refine their algebra skills. When they rely on a CAS to work around it, they make the course a waste of time. Much like a calculus III student I talked with today who is using a CAS to do his homework, fine, but, I wonder how will the test go?
Basic knowledge of graphs of sine, cosine, tangent etc as well as their values is in short supply with many of this generation. I believe this is in no small way tied to overuse of calculators in the precalculus mathematics. On the other hand, it also tied to the fact that their education has been eviscerated by the removal of more and more topics which we deemed too difficult by "educators". Surely, by now it has been clearly demonstrated that the removal of hard topics from elementary school does not bring us a society of like-educated people. In fact, with lower standards we all get left behind because none of us really know the math anymore. I've seen the texts my colleague used as a child. It's sobering to see how far we've allowed the standards to drop.
Hold the line. Teach trigonometric substitution. If nothing else, as an opportunity to teach some much needed trigonometry in calculus.
I should qualify, I do not view university education as a job-training program. I want my students to think. If they want to solve problems without thinking later, that's their business. The reason we solve problems in calculus is not for their applicability. Rather, we solve problems in the interest that the student learn to think. If the problem happens to be really really real-world applicable then great, but, that is not the primary purpose.