Given $C$ a small category (eventually, a small simplicial category) I denote by $UC$ the projective model structure on the category of simplicial presheaves on $C$ as in the paper. Using the kind of argument you have in mind we obtain the following theorem:
Theorem: If $M$ is a simplicial model category, then there is an equivalence of categories between:
- (Simplicial) Functors $C \to M$ taking values in the full subcategory of cofibrant objects.
- Simplicial left Quillen functor $UC \to M$.
In one direction, the equivalence is simply given by restricting to the Yoneda embedding $ C \to UC$ as representable are cofibrant in the projective model structure, this forces the composite functor $C \to UC \to M$ to take values in cofibrant objects. In the converse direction, one takes the unique simplicial left adjoint functor $UC \to M$ and check, using the axiom of simplicial model category for $M$ that this is a left Quillen functor.
However, this is not what the paper you mention proves.
There, they start from a model category $M$ that is not assumed to be a simplicial model category, and a functor $C \to M$ not assumed to takes values in cofibrant objects. And construct a left Quillen functor $UC \to M$ by considering (and choosing) a cofibrant simplicial resolution of the functor $C \to M$ they started from. In particular, the "uniqueness" of the left Quillen functor obtained this way, is only up to homotopy (to be more precise, up to a contractible space of choices).
One abstract way to understand the relation between the two is as follows:
Given $M$ a combinatorial left proper model category, there is a Quillen equivalent simplicial model structure on the category $sM$ on the category of simplicial objects of $M$, (this is explained in the paper "Replacing model categories with simplicial one" by Dugger)
The evaluation at $[0]$ gives a left Quillen equivalence $sM \to M$
One way to understand the non-simplicial theorem is that if you start from $C \to M$, you can see it as a functor $C \to sM$ taking values in constant simplicial objects, then take a levelwise cofibrant replacement to obtain a functor $C \to sM$ taking value in cofibrant object, apply the "simplicial theorem" to get a Quillen functor $UC \to sM$ and finally, post compose with Quillen functor $sM \to M$ that evaluate at $[0]$.
Now for the model structure on $sM$ to exist we need $M$ to be combinatorial and left proper, if you are willing to work with a left semi-model structure instead it is enough to assume that $M$ is an accessible model category (no properness assumption).
But in some sense the central observation of the paper you quote, is that, even if the model structure on $sM$ cannot be constructed, the overall construction make sense with no assumption $M$ (other than being a model category, I guess they also need functorial factorization, I do not remember).
