And indeed, to this day, I don't think there is a published reference for this result.

@Chris I admit I never considered this question (of optimal versus minimal): I tend to work with the first étale cohomology group with coefficients in $\mathbb Z_{p}$ and forget about elliptic curves.

If I understand correctly, the approach of Yamammoto (in the specific case of $G=A_{n}$) is somewhat dual to what you suggest. Instead of killing the ramification by composing with an auxiliary extension, he realizes $G$ as a normal subgroup of a larger group $G'$ and ensures that the ramification can only occur in the quotient $G'/G$. When $G=A_{n}$, this works nicely because $G'=S_{n}$ and it is easy to build polynomials with at most one double roots modulo $p$ for all $p$.

Surely one needs the hypothesis that the splitting field of $x^5+ax+b$ has Galois group $S_{5}$.

I kind of miss the sense of urgency pervasive in the first form of the answer. All this "Will I make it? Will my children be on time for school?" feel was rather enjoyable. That said, the argument about the jumps of the conductor being equal only if $V$ has invariants is nice.

Ben-Ogg's Cor. 1 says that $a(p)=0$ when $e_{p}>1$ and I couldn't see how this was excluded by W.Li (I still can't in fact, but that's my problem, isn't it?).

Like unknown above, I am slightly perplexed by the assertions. When $\pi(f)$ is supercuspidal at $v$, I don't see how to prove this result. In fact, I am already rather surprised by the proof in W.Li' article. The results you quote is proved on page 295 but the proof uses a corollary of Ogg. However, I can't see how the hypotheses of the corollary of Ogg are verified. If you are happy to assume that $\pi(f)_{v}$ is not supercuspidal, then this should follow from the description of the Langlands $L$-factor at $v$.

And if it is not totally real, by the stability of the Hecke algebra under the Rosatti involution and its positivity, the field generated by the coefficients of $f$ is a CM field, isn't it?