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.

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, ...").

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.

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)

2.

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 $F(f\circ g)$.

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 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.

2.

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")

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, ...").