An application of Grothendieck's version of Hensel's Lemma

Assume $$R$$ is a Henselian local ring with $$k=R/m_R$$ ($$m_R$$ is the unique maximal ideal of $$R$$) and $$G$$ a finite flat group scheme over $$R$$. We denote by $$G_k= G \otimes_R k$$ the closed fiber.

There is a well known result that states that there exist an exact sequence of finite flat group schemes over $$R$$: $$\tag{*} 1 \to G^0 \to G \to G^{\text{ét}} \to 1$$ where $$G^0$$ is the neutral component of $$G$$ (= component of "neutral element" $$e \in G$$) and $$G^{\text{ét}}$$ is finite étale over $$R$$.

For such $$G$$ Grothendieck's version of Hensel's Lemma holds (EGA IV 4: 18.5.19): $$\{\text{connected components of G}\}= \{\text{connected components of G_k}\}.$$ I'm familar with some "conventional" ways to prove the existence of the sequence $$1 \to G^0 \to G \to G^{\text{ét}} \to 1$$ which all make use of "classical" formulation of Hensel's lemma (the lifting roots of polynomials $$\bar{f} \in k[x]$$ to those of $$f \in R[x]$$).

I'm really curious if there exist a proof that is more "geometric" proof in the sense that it uses Grothendieck's version of Hensel's Lemma to show the existence & exactness of sequence above.