No one knows. Or at least, no one knows why we know.

I do not mean this flippantly. If you study the proofs of e.g. local and global class field theory (global especially), they use over and over again all kinds of tricks in the yoga of group cohomology to reduce everything down to understanding things we can do by hand, like cyclotomic extensions. But I think this style of proof is very far from a satisfying "why", and I have heard the same opinion from other people (Tate, Rosen). The most satisfying proof in class field theory for me is the Lubin-Tate construction of totally ramified extensions of local fields, precisely because you can make canonical choices and it's reasonably explicit.

Likewise, the Taylor-Wiles method, while an extremely beautiful and powerful idea, is ultimately unsatisfying (to me) as a reason for why Hecke algebras should match deformation rings. If you read the "context-free version" in Section 2 of Diamond's paper "The Taylor-Wiles method and multiplicity one", you'll notice that a subsequence of (quotients of) the auxiliary modules $H_n$ is chosen using compactness. Compactness! Roughly speaking, this corresponds to controlling the relation between deformation rings and Hecke rings at some fixed level by smooshing together a bunch of modular forms at sporadic higher levels, which are chosen in some gratuitously noncanonical way.

I agree with Emerton that converse theorems provide an extremely persuasive reason for believing modularity results. But I also believe that the ultimately "correct" method of proof has not surfaced yet, and who knows how many decades or centuries until it does?

(Let me stress that these are simply my opinions, and nothing more. But they are not unconsidered.)

No one knows. Or at least, no one knows why we know.

I do not mean this flippantly. If you study the proofs of e.g. local and global class field theory (global especially), they use over and over again all kinds of tricks in the yoga of group cohomology to reduce everything down to understanding cases we can do by hand, like cyclotomic extensions of $\mathbf{Q}$ and Kummer extensions more generally. But I think this style of proof is very far from a satisfying "why", and I have heard the same opinion from other people (Tate, Rosen). The most satisfying proof in class field theory for me is the Lubin-Tate construction of totally ramified extensions of local fields, precisely because you can make canonical choices and it's reasonably explicit.

Likewise, the Taylor-Wiles method, while an extremely beautiful and powerful idea, is ultimately unsatisfying (to me) as a reason for why Hecke algebras should match deformation rings. If you read the "context-free version" in Section 2 of Diamond's paper "The Taylor-Wiles method and multiplicity one", you'll notice that a subsequence of (quotients of) the auxiliary modules $H_n$ is chosen using compactness. Compactness! Roughly speaking, this corresponds to controlling the relation between deformation rings and Hecke rings at some fixed level by smooshing together a bunch of modular forms at sporadic higher levels, which are chosen in some gratuitously noncanonical way.

I agree with Emerton that converse theorems provide an extremely persuasive reason for believing modularity results. But I also believe that the ultimately "correct" method of proof has not surfaced yet, and who knows how many decades or centuries until it does?

(Let me stress that these are simply my opinions, and nothing more. But they are not unconsidered.)