It is certainly possible to define the Jones polynomial from the skein relation. One point to understand, though, is that showing that the relation defines a polynomial is not a matter of showing invariance under Reidemeister moves since the skein relation is not a relation on diagrams, but of links themselves. To explain this better, consider the following formalism:
Suppose $\mathcal{K}$ is the set of all oriented links in $S^3$ up to isotopy, and let $\mathbb{Z}[t^{\pm 1/2}](\mathcal{K})$ denote the set of $\mathbb{Z}[t^{\pm 1/2}]$-linear combinations of isotopy classes of links. Suppose $L_+,L_-,L_0$ are links that outside a given ball are the same, and within the ball the three links each have the respective $2$-tangles you illustrate. Let $M$ be the submodule generated by $\text{unknot}-1$ and by $t^{-1}L_+-tL_--(t^{1/2}-t^{-1/2})L_0$ for all such triples of links. The claim is that the skein module $\mathbb{Z}[t^{\pm 1/2}](\mathcal{K})/M$ is isomorphic to $\mathbb{Z}[t^{\pm 1/2}]$. Then, $V(L)$ is defined to be the image of $L$ in this quotient.
It is not difficult to show that the skein relation is enough to represent each link as a polynomial in this quotient. The first step is to show that a split union of $n$ unknots is represented by $(-t^{1/2}-t^{-1/2})^{n-1}$. The second is to note that every link has finite unlinking number, and then be careful to show that the skein relation allows one to reduce the link to a polynomial. (This is, every link $L$ has at least one $p\in\mathbb{Z}[t^{\pm 1}]$ such that $L-p\in M$.)
The more difficult part is to show that there aren't more relations in $M$. It could very well be that the quotient will be some quotient of $\mathbb{Z}[t^{\pm 1/2}]$. Lickorish gives a proof of the well-definedness of the HOMFLY polynomial (and hence the Jones polynomial) in Introduction to Knot Theory (chapter 15) using the following approach. Let $D_n$ be the set of link diagrams with at most $n$ crossings, and inductively prove that the polynomial associated with links in $D_n$ are unchanged by Reidemeister moves (along with a "Reidemeister IV": passing an unknot under a strand) that involve no more than $n$ crossings, and prove that "ascending diagrams" have the polynomial associated to a disjoint union of unknots. Once this is done, the idea is that any sequence of Reidemeister moves between two equivalent links $L$ and $L'$ can be conducted within $D_n$ for some $n$, hence there are no additional relations.
As you note in the comments, the Kauffman bracket approach for the existence of the Jones polynomial is much easier. The conceptual benefit of a skein relation is that it does not involve diagrams.
