I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following :
Let $K$ be an algebraically closed field. A connected (affine) algebraic subgroup of $(K,+)^n$ having dimension $1$ is isomorphic to $(K,+)$.
Any ideas for elementary proofs ?
What you stated is technically wrong since for example one can take $\mathbb{G}_a\times \alpha_p$ over $\overline{\mathbb{F}_p}$. So, you probably want to assume that the subgroup is also reduced. If so, then see the following.
If you're curious about a not-so-elementary proof of the classification of smooth connected one-dimensional algebraic groups you can see the following two links (NB: full disclosure, this is my blog):
For a sketch of the idea of this proof you can see my explanation here: Classification of the algebraic affine smooth group schemes of dimension 1.
The claim then follows, as Ariyan noted, since any subgroup of $\mathbb{G}_a^n$ must be unipotent and thus must be $\mathbb{G}_a$ by the above classification.
If one wants another proof, possibly not so elementary (but in a different way), proof one can see [Mil, Corollary 14.33] and [Mil, Corollary 14.53].