I found a chapter from Goodman-Wallach's book (pp. 186-189) and Sengupta's book (pp. 249, 257) that seem to use double commutants to decompose representations. The notation and terminology are over my head and I am not sure if they are decomposing $T$ or $\mathbb{C}[T]$, but is this what you have in mind? Maybe I can decipher them.

Thank you for the elaboration. It is just that I am more familiar with the harmonic analysis side of the representation theory that uses $T(g)$ and equivariant maps explicitly, and not so much with modules and ideals, but I'll try to work through your proof. Would the double commutant theorem give a proof that does not require using the decomposition of $\mathbb{C}[G]$? $M(T)$ is the subspace of $L^2(G)$ spanned by the matrix elements $T_{ij}(g)$ in some basis, quoted here.

@YemonChoi I am mostly interested in finite groups, compact at most, as in the standard formulations of the Peter-Weyl theorem. And how it transfers to similar representational algebras that are not explicitly made of functions.

Finite $G$ is all good, but I am having hard time translating between the language of semisimple algebras and modules and group representations and matrix elements. What does "modules break up", etc., mean in the latter? Any reference that does the translating? Is it true that $\mathbb{C}[T]\simeq M(T)$ or, perhaps, $M(T^*)$, and if so how does one 'see' it? Can this be proved without decomposing $\mathbb{C}[G]$ first so that it follows as a special case? There was a question about double commutant for representations, but alas, no answer.

"Burnside's theorem" by itself leads to different results. The correct search term seems to be "Burnside's theorem on matrix algebras". An elementary proof based on rank 1 matrices, which goes back to Halperin and Rosenthal, is in The simplest proof of Burnside's theorem on matrix algebras.