Is this explanation of Lubin–Tate theory as a generalization of roots of unity mathematically correct?

I am preparing a presentation and would appreciate feedback on the following explanation connecting the multiplicative group with Lubin–Tate theory.

Let $K$ be a field and consider elements $x,y\in K^\times$ near the identity $1$. Write

$$ x=1+X,\qquad y=1+Y. $$

Then the group law on $K^\times$ becomes

$$ xy=(1+X)(1+Y) =1+X+Y+XY. $$

Thus, if we define

$$ F(X,Y)=X+Y+XY, $$

we obtain the formal multiplicative group law.

The multiplication-by-$n$ map $ [n](X)=(1+X)^n-1$ is an endomorphism of $F$ as it satisfies $[n](F(X,Y))=F([n]X, [n]Y)$

The $n$-torsion points of $F$ are the zeros of $[n](X)$, namely the solutions of

$$ (1+X)^n=1. $$

Hence

$$ X=\zeta_n-1, $$

where $\zeta_n$ is an $n$-th root of unity. Therefore

$$ F[n]=\{\zeta_n-1:\zeta_n^n=1\}, $$

and

$$ K(F[n])=K(\zeta_n-1)=K(\zeta_n)=K(\mu_n). $$

Thus adjoining the $n$-torsion points of the formal multiplicative group is equivalent to adjoining the $n$-th roots of unity.

My interpretation is that Lubin-Tate theory generalizes this phenomenon: one replaces the formal multiplicative group by a Lubin-Tate formal group $F$ over $\mathcal O_K$, and replaces roots of unity by the torsion points of $F$. Adjoining these torsion points then generates the abelian extensions appearing in local class field theory.

Question. Is this interpretation mathematically accurate? In particular, is it reasonable to view Lubin--Tate theory as a generalization of the fact that roots of unity arise as torsion points of the formal multiplicative group?

Ofcourse here $K$ should be a local field.

I am asking this because a broad mathematical audience will be there, where many attendees will not be specialists in algebraic number theory or local class field theory.