A short proof of $(*)$ uses the characterization of $0^\sharp$ in terms of mice. A nice account of this version is in Ernest Schimmerling's paper "The ABC of mice".

The idea is that a sharp is a kind of "local measurable cardinal". More formally, we can think of a set equivalent to $0^\sharp$ (in a strong sense) to be of the form $M=(L_{(\kappa^+)^L},\in,{\mathcal U})$ for some $L$-cardinal $\kappa$ and some $L$-measure ${\mathcal U}$ on $\kappa$. We ask some definability requirements of $M$ that imply that $M$ is countable (in $V$), and we require that we can iterate ultrapowers by ${\mathcal U}$ and obtain well-founded models.

(I say "local measurable" because as soon as we continue the constructible hierarchy over $M$, we are able to define a map from $\omega$ onto $L_{(\kappa^+)^L}$, so not only is $M$ countable, but ${\mathcal U}$ is very far from being a measure in $L[M]$. On the other hand, ${\mathcal U}$ measures all the subsets of $\kappa$ in $L$, so ultrapowers using ${\mathcal U}$ of $L$ or initial segments of $L$ make sense.)

It follows at once from continuity properties of embeddings that if $\alpha<\beta$ are uncountable cardinals in $V$, then iterating the ultrapower embedding by ${\mathcal U}$ will send $L_\kappa$ eventually to $L_\alpha$ and later to $L_\beta$, and so in particular $L_\alpha\equiv L_\beta$.

What we are doing here is casting Kunen's characterization of $0^\sharp$ as the key defining property. All of this, and the way to recover the usual version of $0^\sharp$ from this one, is explained in very nice detail in Ernest paper, that I recommend you study.

The presentation itself is part of the folklore of the subject. I learned it from John Steel. It is essential to understand this presentation if one is to make sense of current fine structure or inner model theory.

That being said, the EM-blueprints approach is not outdated. It is essentially the only way we can make sense of sharps of sets that are not 'internally well-ordered'. This idea was first developed by Robert Solovay when he introduced ${\mathbb R}^\sharp$ in the course of his work on determinacy. It has been much refined by Hugh Woodin during his study of the Chang model.