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 cover up certain - more precisely the Abelian totally ramified - 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.
( to make it more precise, jump to next paragraph or read @KConrad's comment below)

But note that the Lubin-Tate theory is not a gadget to cover up all finite extension of a local field, but only the totally ramified extensions (for $\Bbb Q_p$ that would the $p$-part) via adjoining certain torsion points of certain $1$-dim formal group to obtain family (... more precisely a tower) of certain extensions $K_{\pi, n}$, which contain totally unramified Ab. fin. extensions. And the pun is:

It does it exhaustively for the totally ramified part in the sense that for every fin Ab. totally ramified ext'n $L/ \Bbb Q_p $ there exist a $K_{\pi, n((L)}$ obtained by adjoining appropriatetorsion points of the associated $1$-dim formal group to $\Bbb Q_p$ which contains $L$. And in such way all finite totally ramified extensions can be exhausted, exactly in same vein - of course only on level of analogy - as in Weber's Theorem the cyclotomic extensions exhaust fin Abelian extensions of $\Bbb Q$. So one could phrase it as slogan that Lubin-Tate theory is exhaustive wrt the totally ramified part.

Rmk: The extensions $K_{\pi, n}$ are called non surprisingly the Lubin-Tate extensions.

Now comming back to the analogy to the torsion points of elliptic curve.
Question: How "exhaustive" - in admitedly rather vague sense above - is the procedure of 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 covered up as subextension exactly in same vein as in Lubin-Tate, resp Weber's result by Lubin-Tate, resp. cyclotomic extensions.

As below comments suggest, it's highly too optimistic to expect that with this method really all finite Abelian extensions be exhaused like in case of Weber's theorem for Abelian extensions of $\Bbb Q$. But can one say on what kind of Abelian extensions of $K$ can be exhaused by this torsion points construction? Do these share some number theoretic common properties?

Subsequent question: There are some qualitative results for elliptic curves with CM or without it, and how goes CM into it?

Say, what fails precisely if the consided elliptic curve would not have CM? (ie the endomorphism ring equals $\Bbb Z$) What would exactly break down?

Also it appears that to have gain some qualitative results one resricts to a rather "narrow" class of fields& extensions, eg totally imaginary quadratic extensions of totally real fields (compare with a comment below).