$0^\sharp$ and transcendence over $L$

Question

I've started to study the $0^\sharp$ principle from Kanamori's The Higher Infinite, and there are a few interesting, yet a bit vague, remarks scattered across the text that intrigue me and whose core meaning I'd like to understand. I'll write them in the order of appearance.

After having made and introduction about indiscernibles (pag. 101) he says

Silver realized that because of the uniformity of the constructible hierarchy $L$, if a sufficiently rich structure had enough ordinal indiscernibles, the theory of the structure fueled by the class of ordinals in the role of indiscernibles can be used to generate $L$. This harkens back to Gödel's original impredicative use of the class of ordinals to construct $L$ by extending type theory, but the indiscernible generation shifts the weight of the construction squarely on the theory and leads to striking consequences about the distance between $V$ and $L$.

A few questions:

1. The theory he is referring to is the EM blueprint, right? In which sense it can be used to generate $L$? Is he referring just to the fact that, if such rich structure exists, $L$ can be recovered as the Skolem hull of a class of indiscernibles?
2. I don't understand the link he makes with Gödel's definition of the constructible transfinite hierarchy. To me, Gödel's generation of $L$ is a "true" generation , that uses only the transfinite as a given "impredicative generating principle". On the other hand $0^\sharp$ gives a very powerful description of $L$'s structure, but I don't see how this "indiscernible generation" can be regarded as a generation (whathever it means), and as a consequence I don't see the point of the comparison.

Then at the end of the section (pag. 111) he says

Kunen soon showed that [...]: "If there is an elementary embedding $j:L\rightarrow L$ then $0^\sharp$ exists" [...] Then in 1974 Jensen established his celebrated Covering Theorem: "$0^\sharp$ does not exists iff for any uncountable set $X\subseteq \text{On}$ there is a $Y\in L$ such that $Y\supseteq X$ and $|X| = |Y|$." [...] This results have buttressed the existence of $0^\sharp$ as the focal hypothesis of transcendence over $L$.

1. Having said that $0^\sharp$ is sort-of "canonical" transcendence principle over $L$, my question is what makes in general a good transcendence principle (over $L$ or over a generic inner model), besides being resonably weak? Should it give us a strong "inner structure" of the model it transcendes? Should it gives us a strong "outer structure" of the outer universe, for example by constraining the size of $V$ in its absence (as the Covering Theorem does for $0^\sharp$)?
2. Are there interesting\powerful consequences of the Covering Theorem that strengthen further the intuition that in the absence of $0^\sharp$ the universe $V$ needs to be very close to $L$? And are there, on the other hand, interesting results\remarks that strengthen further the intuition that in the prensence of $0^\sharp$ the universe $V$ needs to be very far to $L$?

Thanks!