This question is very broad, and its answers subjective (how do you quantify usefulness?), but I can give some comments.
1. Asymptotic convergence is not broadly guaranteed a priori.
When solving an elliptic PDE using the finite element method, there are well-established results such as the Lax-Milgram theorem and Cea's lemma that furnish straightforward "guarantees" of solving efficiency for such a problem. This is nice because the validity of such theorems depends only on the nature of the equation, not of the solution.
Compare that with, say, trying to obtain a Taylor series representation of a PDE solution over some expansion point. Who says the function is analytic over the whole domain? And estimating the radius of convergence with something like the Cauchy-Hadamard theorem requires informed guesses as to the behavior of the coefficients of such a series, which are themselves obtained from solution attempts.
You often get the same problem in perturbation theory; who says that a matched boundary layer approach or WKB approach will yield a convergent series? Often perturbation expansions of this type rely on constructing a sequence of linear operator equations out of a single nonlinear operator equation, and hoping that your sequence of equations has a solution up to some order (as determined by the Fredholm alternative, or other such theorems). This involves getting rid of terms that are incompatible with these operators as they appear in the solution sequences, which is a fairly unpredictable process. As I understand it, there are very few theorems that guarantee convergence of such series without having some information about the solution.
2. Asymptotic methods are fundamentally local.
Series approaches involving orthogonal function sets (Fourier series, Chebyshev polynomials) use the geometry of Lebesgue spaces to make guarantees about the "goodness" of the approximation: for example, that the $L^2$ norm of the residual always decreases as you increase the terms in the series, that the series is the best representation of the function in a specific subspace, etc. This is due to the fact that such series are approximating the function globally, whereas asymptotic methods approach solutions locally. As a result, there are few such broad, general guarantees in asymptotic series, which can show weird divergent behaviors in specific regions of the domain as terms in the approximation increase even if it behaves perfectly well near the point of local expansion.
The strange behaviors of asymptotic series are, for me, epitomized by Carrier's rule: "Divergent series converge faster than convergent series because they don't have to converge".
3. It is difficult to predict the computational efficiency and solvability of algorithmic asymptotic methods.
Because asymptotic methods are local and have few a priori guarantees of behaving "well", it is very difficult to predict whether an algorithm employing such methods terminates or, if it does, how quickly it would do so. Consider, in perhaps a naive way, an algorithm attempting an asymptotic expansion that must match some boundary condition in a region far from the expansion point, and the algorithm terminates when the approximation is within some error margin. It may well be the case that no such expansion of the type you want exists (à la Stokes or Whitehead paradox), and as a result your algorithm keeps fruitlessly searching for the right function ad infinitum.