Here are some leads for 1. and 2. (this is far from a full answer, but slightly too long for a comment). I only went in the "disadvantages of this approach" direction, too so that's another reason it's not a full answer. 

1. 

a- A mathematical advantage is that when you go to higher category theory/homotopy theory, then composition in the shape *does* matter (you need the naturality square for $f,g$ and $f\circ g$ to be compatible, and so on for higher morphisms). So in a sense what you describe is the consequence of the fact that in 1-category theory, equality of morphisms is a property and not extra structure. 
In particular, with the definition "shapes as categories", the higher generalization is clearer. This is to me the main disadvantage of that approach, especially in this day and age. 

b- Another advantage is that it allows for a unified framework (categories), where you don't have to introduce a new notion ("the category of diagrams with shape a directed multigraph") for something that already exists and is useful for many reasons (functor categories) - I guess this is somewhat the other side of the coin for your point 4. 

c- Related to that is Zhen  Lin's comment, that things indexed by categories somehow strictly generalize things indexed by directed multigraphs precisely because of your point 1 - this is also related to the previous point, that the category of diagrams is a special case of a functor category, but not the other way around (note that *co/limits* are not generalized that way, but categories of diagrams *are*)

d- Finally, a last advantage that I can see (related to your point about Kan extensions) is that many of the diagrams you naturally encounter - except for the "standard diagrams" : $\mathbb N$, co/products, co/equalizers, pullbacks/pushouts - are "naturally" indexed by categories. 
That is, often when you encounter a co/limit, it'll be because you had a functor in your hand, and wanted to understand something about it.

Here are some examples of this phenomenon: 

i. In all questions of "generation by colimits" (e.g. the fact that presheaves are canonical colimits of representable presheaves), you get functors that are "naturally" indexed by slice categories $S_{/x} = S\times_C C_{/x}$ for some $S$ with a functor to $C$. Oftentimes, the properties of the *category* $S_{/x}$ are important to get some information out of that, or about that : is it filtered, is it sifted, weakly contractible, ... ? This relates of course to your example about Kan extensions, but comes up often in practice (e.g. if $F$ is a finite limit preserving presheaf on $C$, then $C_{/F}$ will be filtered, which allows for the multiple characterizations of the ind-completion)

ii. Many objects are defined as functors on some (non free) category, and you often want to take their colimit along the indexing category, or something related to it - but this almost always "naturally" comes in the form of a diagram indexed by a category. For a specific example, take $BG$ : functors out of it are objects with a $G$-action, and you'll often be interested in taking orbits or fixed points and stuff like that. 

iii. This is very specific but comes up a lot: oftentimes, you have a construction that produces a functor $\Delta^{op}\to C$ out of some piece of data in $C$ (a monad on $C$, or more specialized, a monoid in $C$, a left and a right module thereon, ...), and whose colimit is very interesting (or the limit if you get a functor from $\Delta$). Again, you could forget the fact that $\Delta^{op}$ is a category, but the diagram you got was "naturally" a functor. Another way to phrase this is that you often want to precompose diagrams (see 2.b- below) with functors, and this might not be possible with graphs, or at the very least more complicated to phrase. 

2. 

 a- Point 1.b- above seems to also be a pedagogical advantage as far as I can tell : you have strictly less things to remember/understand if diagrams are introduced as being indexed by categories. 

b- A certain number of arguments about limits and colimits involves taking composites $F(f)\circ F(g)$ where $cod(g) = dom(f)$, and are then just easier to state if you can reduce to $F(f\circ g)$ (here "easier" may be an overstatement, I should say "less awkward")

For instance, suppose you have an inverse limit indexed by $\mathbb N$ and want to replace it with the one indexed by $2\mathbb N$ for some cofinality argument - of course you can do the construction by hand, but it's easier to just define a functor $2\mathbb N\to \mathbb N$ than a directed multigraph morphism (which is a notion you have to introduce too anyways - this comes back to point 2.a- above). 

But cofinality arguments aren't even the only ones where you want to something like this (to give a second example, a condition you often meet is "for every $f$, $F(f)$ is blah" for some interesting notion "blah", which you would have to replace with "for every composable sequence, ..."). 

Here are some examples: 

i. Let me stress the cofinality example, which is especially important in higher category theory where you can less often compute co/limits by hand, and so cofinality arguments come in very handy. 

ii. When you're working with directed colimits in concrete categories, it's just less awkward, in the situation $i\leq j \leq k$, to have the composite be part of the diagram rather than have to say "well it works because $F(ij)$ and $F(jk)$ are in the diagram"

iii. The Mittag-Leffler argument, when you're working with $\mathbb N$-indexed diagrams in suitably nice abelian categories (even very concrete), is always very nice, and it's just less awkward to write "the sequence $\mathrm{im}(f_{ki})$ stabilizes" than "the sequence $\mathrm{im}(f_{k(k-1)}\circ f_{(k-1)(k-2)}...f_{(i+1)i})$ stabilizes". (this is secretly a cofinality argument, but with a different feel so let me add it anyways, because this part is about pedagogy)