the universal representation for c*-algebras is defined as the direct sum over all GNS-representations $(H_\phi,\pi_\phi)$ where $\phi$ is a state on the c*-algebra. This representation is faithful.
My question: Couldn't one also form the direct sum of all GNS-representations $(H_\phi,\pi_\phi)$ where $\phi$ is just a positive linear functional and thus still obtain a faithful representation?
And if so: What's the benefit of not doing it that way?
On
To add to what Aweygan said, even using all the states is too much, in a sense. If any two states $\psi,\phi$ are unitarily equivalent, then $(H_\psi,\pi_\psi)$ and $(H_\phi,\pi_\phi)$ are unitarily equivalent, thus the two of them do not contribute more information than just one of them.
Because of the above, it is customary to add not over all states, but over their unitary equivalence classes.
For your first question, yes, taking the direct sum over all positive linear functionals does yield a faithful representation. All we need is that for each $a\in A$, there is some positive functional (or state) $\phi$ such that $\phi(a)\neq0$, and that's certainly true if we add more positive linear functionals.
Note that if $\phi,\psi$ are positive linear functionals with $\phi=\lambda\psi$ for some $\lambda\in\mathbb R_+$, then $\pi_\phi=\pi_\psi$. Thus we're not losing any information by restricting to states (and the zero functional adds nothing). And if you're lazy like me, you want to get as much as possible while doing as little as possible.