What is the theorem of the highest weight used for? $\DeclareMathOperator\End{End}$Over the past few months, I have taught myself the classification of reductive groups, and continued to non-abelian (as well as a small venture to non-compact) Harmonic Analysis.
I am now trying to put everything that I learn together into something coherent. Let's, therefore, take the case of compact real Lie groups. By Chevalley the category of compact real Lie groups is equivalent to the category of $\mathbb{R}$-anisotropic reductive linear algebraic groups whose connected components have $\mathbb{R}$-points. In particular, all of the representations of a compact real Lie group are algebraic.
Therefore, by the Theorem of the Highest Weight, we have a classification of the unitary dual (the set of irreducible unitary representations) of a real compact Lie group $G$.
From the Harmonic Analysis perspective:
$$L^2(G)\cong\bigoplus_{\pi\in\hat{G}} \End(\pi)(\cong \bigoplus_{\pi\in\hat{G}} \pi\otimes\pi^*).$$
It remains, therefore, to choose a basis of each $\End(\pi)$, for each $\pi$ guaranteed by the Theorem of the Highest Weight.
I looked at the examples in Folland's book on Harmonic Analysis, and did not see any mention of the Theorem of the Highest Weight. This seemed to have been done completely ad hoc.
I was also told in an old question of mine to take a look at Zonal Spherical Functions, in the particular case that $K=1$. I must confess, I find Zonal Spherical Functions to be quite confusing. I suspect that the point is that these are the method by which Harish Chandra proved the Plancherel Theorem (the non-compact variant of Peter-Weyl), but it's not clear to me how to use this in practice.
But either way, I end up wondering: what is the point of the Theorem of the Highest Weight? Does it provide a basis for each $\End(\pi)$? If it does, then what is it? If it doesn't, and we end up using some other method for finding the basis for each $\End(\pi)$ (a method that appears to ad hoc show that we have exhausted all of the irreducible unitary representations), then what good is the Theorem of the Highest Weight?
To put it succinctly: what utility does the Theorem of the Highest Weight provide, and how, if at all, does it fit into the picture of finding a basis for each $\End(\pi)$?
 A: $\DeclareMathOperator\SL{SL}$One direct response to the question of "what does the theorem of the highest weight give us?" is that the highest weight completely determines the eigenvalue by which Casimir acts on that irreducible. (And all of the center of the universal enveloping algebra, as well ….) (This illustrates a semi-tangible form of Harish-Chandra's isomorphism describing the structure of that center.)
The highest weight does also approximately (to my mind) tell how to find a spanning set for the irreds (in general). In small cases it can give a basis, but I think in general the specification of a basis is significantly subtler, … keywords "crystal basis" …. Names include Gelfand, Kashiwara, Lusztig, et al., and relatively recent results (perhaps showing my limited awareness here) from Brubaker, Bump, Friedberg … Chinta, Gunnells, … et al.
One "disappointing" result I do remember is that the ideal structure in Verma modules turns out to be more complicated than the most optimistic conjecture. So it is non-trivial to describe the linear dependence of images of the highest weight vector under lowering operators … unfortunately. For $\SL_2$ it certainly turns out well, and I think for $\SL_3$, but I believe already for $\SL_4$ there is a (complicated) counter-example ….
EDIT: to clarify, e.g., as prompted by @LSpice's remark, … given a choice of positive and negative roots, any repn (finite dimensional or not) with a (unique) highest weight vector is spanned by all the images of that vector under the "lowering operators", that is, the operators in the universal enveloping algebra coming from the negative root-spaces in the Lie algebra.
For $\SL_2$, it is easy and standard to see that the images of the highest weight vector under the (essentially unique) lowering operator are a basis … with the eventually-too-lower image being $0$. (The lowest weight vector ….) There are no non-obvious relations.
Even with $\SL_3$, it is non-trivial to see that the "reasonable" relations among images under negative root-space (lowering) operators are all there are. This amounts to something like showing that the only submodules of a Verma modular with given highest weight (Verma module being the universal module with given highest weight) are again Verma modules. This optimistic idea proves tooooo optimistic, with entailed complications.
EDIT_2: Jacques Dixmier's "Enveloping Algebras" (original in French, too) I believe gives a citation for the failure of submodules of Verma modules to be isomorphic to Verma modules, for $SL_4$.
A: The theorem of the highest weight tells you the structure of all finite-dimensional representations of simple (complex) Lie groups. In particular, it tells you what is the set $\widehat{G}$ that you are summing over. For some purposes this itself might be enough, i.e. you get that $L^2(G)$ is a completion of direct sum and so each $L^2$ function can be expressed as a series of some basis functions. Compare with the prototype case of $L^2(S^1)$ with Fourier series vs $L^2(\mathbb{R})$ with Fourier transform.
If you ask what are actually these basis functions, then the highest weight theorem will help you as long as it comes with a reasonable construction of irreducible representations, since the matrix coefficients
$$
\{m_{u,v}: g \mapsto \langle g u | v \rangle \,\big|\, u, v\in \mathbb{V}, \mathbb{V}\in\widehat{G}\}
$$
form a basis of $L^2(G).$
Sure, you can do some ad hoc construction and never really talk about the group structure. In the same way you can talk about Fourier series and never mention that those exponentials there are actually group characters of $U(1).$ But this perspective allows you to subsume Fourier series, Fourier transform, Discrete Fourier transform and much more in one theory. Similarly, for compact simple Lie groups, the theorem of the highest weight and the Peter-Weyl theorem unlock a sort of recipe how to go about  "Fourier transform" for $L^2(G).$
