When working on a rational problem (over $\mathbb{Q}$), you can't do much analysis - so you lack quite a few tools.
The obvious solution is then to pass to the completion, where you'll be able to do analysis ; so most people go to $\mathbb{R}$. But that isn't that natural : $\mathbb{Q}$ has several completions in fact, and choosing the absolute value to measure distances wasn't the only choice. You could have gone to the $p$-adic numbers too, and use their special properties to gain further insight on your initial problem.
So somehow you gain the idea that perhaps solving your initial problem (which is called global) might involve looking into the various problems obtained by pushing it into the various available completions (which are called local).
That means you gain huge means of study, but now there are two prices to pay : you have the question whether some kind of Hasse principle applies, which is "How equivalent is it to solve the problem locally everywhere and globally?", and you have the problem that you need to work in all of those.
That last problem is where adeles come into the picture. Working adelically means you're effectively working simultaneously in all places -- and you mostly put them on an equal footing. You're still able to go into a single local place if needs arise, but you get an object which puts the pieces together.
There is another nice thing about adeles : if your initial problem wasn't just over $\mathbb{Q}$, but other any other kind of global field (a number field or a function field), then again you'll have a notion of adeles, and most tools will work the same. In fact, getting insight on a problem for a type of global fields and pushing the idea in the adeles is a good way to know what to look for in the other type of global fields. There are problems which are thus solved for some of them, and conjectures for others, precisely for this reason.
So I think what I wrote makes clear why wanting to replace adeles by just $\mathbb{R}$, while it will give some understanding of things (in a single place!), will also pretty much destroy the whole symmetry of the matter, and hence lose much of it.
I would disagree with your last two points just as wccanard does in his comment: automorphicity of $L$-functions is part of global Langlands functoriality, not the local conjectures (although the two are related).
I would also disagree with your third point: instead of nonabelian harmonic analysis, it is automorphicity that translates into functional equations and vice versa. For example, by the automorphicity of certain Eisenstein series we can see that certain $L$-functions coming from automorphic forms satisfy a functional equation, and from this we can sometimes deduce that the $L$-function is itself automorphic (not just pretends to be).
Regarding nonabelian harmonic analysis I would say that it naturally leads to the notion of automorphic forms and representations (via the spectral decomposition), and it also provides a good framework to study them (including establishing certain cases of functoriality without using $L$-functions).
Best Answer
For simplicity assume that $G$ is a reductive $\mathbb{Q}$-group that is anisotropic. Assume that it admits an automorphism $\theta$. Let $f \in C_c^\infty(G(\mathbb{A}))$.
One has the usual kernel $$ K_f(x,y):=\sum_{\gamma \in G(F)}f(x^{-1}\gamma y). $$ The trace formula is an expression for $$ \mathrm{tr}R(f):=\int_{G(\mathbb{Q}) \backslash G(\mathbb{A})}K_f(x,x)dx $$ The twisted trace formula is an expression for $$ \mathrm{tr} R(\theta^{-1}\circ f):=\int_{G(\mathbb{Q}) \backslash G(\mathbb{A})}K_f(x,{}^{\theta}x)dx. $$ The point is that if one expands $\mathrm{tr} R(\theta^{-1} \circ f)$ in terms of cuspidal automorphic representations, then only those representations that are isomorphic to their $\theta$ conjugates contribute, for the same reason that the trace of an $n \times n$ matrix associated to a cyclic permutation of order $n$ vanishes. Thus the twisted trace formula isolates only part of the spectrum. One then tries to compare this part of the spectrum to spectra of other groups and thereby deduce cases of Langlands functoriality. This is the ultimate goal of what is known as twisted endoscopy.
The morning seminar (arXiv:1204.2888), as reproduced by Labesse and Waldspurger, is a high-level reference for this.