As far as I know, Tate's thesis has been successfully applied in two fronts:

Hecke L-functions, by Tate and Iwasawa (and Teichmüller, Witt, Schmid)

Automorphic L-functions, by Jacquet, Shalika, Shapiro etc.

Of course the first approach works for everything below Hecke in the L-function food chain (Dedekind, Dirichlet, Riemann).

My question is, given that the same ideas work for those extremes, abelian on the one side and... well, supposedly all of them on the other, why hasn't been any progress using harmonic analysis to study Artin L-functions?

For example, Tate's thesis for 1-dimensional Artin representations is just... Tate's thesis again (this is, using $W_k$ instead of Galois). This detail is obvious enough, but I haven't seen the details ever worked out in any detailed (I don't think I've seen it mentioned at all).

If the analysis goes through for general automorphic objects, I can't imagine the problem being in dealing with noncommutative locally compact groups. So, where is the problem exactly?