I am looking for a proof of the fact that every faithful representation of $\mathrm{Spin}(V)$ can be realized as a restriction of Clifford module.
Realizing the spin group $\mathrm{Spin}(V)$ as a subset of the Clifford algebra $\operatorname{Cl}(V)$, one can easily see that a $\operatorname{Cl}(V)$-module $W$ defines a faithful representation of $\mathrm{Spin}(V)$, since the relation $(-id) + id = 0 $ in the Clifford algebra implies that the elements $-id, id\in \mathrm{Spin}(V)$ map to distinct elements in $\operatorname{End}(W)$. Equivalently, the group homomorphism
$$ \mathrm{Spin}(V) \to \mathrm{GL}(W)$$
does not factor through $\mathrm{SO}(V)$.
In fact, one defines the "spin representation" $\Delta$ as the restriction of irreducible $\operatorname{Cl}_0(V)$-module onto $\mathrm{Spin}(V)$; see this MSE question. However, how can one conclude that such restrictions exhaust all faithful irreducible representations of $\mathrm{Spin}(V)$?
One way to this end would be taking E. Cartan's approach and identifying the highest weights of $\mathrm{Spin}(V)$-modules.
However, is there a more "direct" way to prove the claim? Namely, is there a known proof of $$ `` \frac{\mathbb{C}[\mathrm{Spin}(V)]}{((-id) + id)} \text{-module} = \operatorname{Cl}_0(V)\text{-module} " $$
that does not depend on weight theory?
