According to a remark from wikipedia the motivation of Lubin-Tate theory arose from the analogy to the way in which elliptic curves $E/K$ over a number field $K$ with extra endomorphisms (ie those posessing complex multiplication CM) are used to give abelian extensions.

From level of exhaustivity (see below what I mean by this precisely), how "close" is this analogy? The loose connection is that in both cases we use the torsion points of an "auxilary object" (some $1$-dim formal group in LT-theory, resp a elliptic curve with CM ) to construct certain exitensions of the given local field - feel free to think of $p$-adics $\Bbb Q_p$ -, resp. Abelian extensions of the base field $K$ of elliptic curve $E$ choosen to be a number field.

But note that the Lubin-Tate theory is not a gadget to produce explicitly all finite extension of a local field, it is a gadget to construct the ramified extensions (for $\Bbb Q_p$ that would the $p$-part) explicitly via adding appropriate torsion points of the formal group (what was previously not possible to do explicitly)

Note that it does it exhaustively for the ramified part in the sense that it generates in that way all finite totally ramified extensions. So Lubin-Tate theory is exhaustive wrt the ramified part.

More generally, the extensions by means of Lubin-Tate theory are called - surprise - Lubin-Tate extensions.

Now comming back to the analogy to the torsion points of elliptic curve.
Question: How "exhaustive" is the procedure to generating Abelian field extensions of the function field $K$ of its elliptic curve via the torsion points? (practically, one adjoints the coordinate entries of the considered torsion points to the base number field)

Ie, which finite Abelian extensions can be generated via this method, so which are the analoga of the Lubin-Tate extensions?

Could with this method really all finite Abelian extensions be exhaused like in case of Weber's theorem for Abelian extensions of $\Bbb Q$?
If not, what kind of Abelian extensions of $K$ can be generated by this torsion points method? Do these share some number theoretic common properties?

Metaquestion: What fails in the constrction precisely if the elliptic curve would not have CM? (ie the endomorphism ring equals $\Bbb Z$)