In the context of Goodwillie's paper, he's got an explicit natural transformation $f:holim_I(X)\to holim_J(X|_J)$, where $X:I\to Top$ is a functor to spaces, and $J\subset I$ is a subcategory of $I$. With the construction of holim he's using, this map is always a fibration.
What if you tried to use a different construction of holim? Then maybe you get a map $f'$ which is not a fibration anymore. In that case, you could still have taken the homotopy fiber of $f'$, and this would be a notion which is invariant under weak equivalence. That is, you could (functorially) replace $f'$ with a fibration via the path construction, and take the fiber of that.
Of course, the homotopy fiber is exactly the thing he wants here. In fact, he's manufactured the situation exactly so that the homotopy fiber he wants is just the fiber of this map.
(It's worthwhile to note that in his setting, the category $I$ (which is a cube) has an initial object $\varnothing$. This means that the evident map $holim_I(X)\to X(\varnothing)$ is a weak equivalence. In other words, $holim_I(X)$ is really just $X$ evaluated at $\varnothing$, but modified so that it maps to (and fibers over) $holim_J X|_J$.)
Quillen's original proof (in Homotopical Algebra, LNM 43, Springer, 1967) is purely combinatorial (i.e. does not use topological spaces): he uses the theory of minimal Kan fibrations, the fact that the latter are fiber bundles, as well as the fact that the classifying space of a simplicial group is a Kan complex. This proof has been rewritten several times in the literature: at the end of
S.I. Gelfand and Yu. I. Manin, Methods of Homological Algebra, Springer, 1996
as well as in
A. Joyal and M. Tierney An introduction to simplicial homotopy theory
(I like Joyal and Tierney's reformulation a lot). However, Quillen wrote in his seminal Lecture Notes that he knew another proof of the existence of the model structure on simplicial sets, using Kan's $Ex^\infty$ functor (but does not give any more hints).
A proof (in fact two variants of it) using Kan's $Ex^\infty$ functor is given in my Astérisque 308: the fun part is not that much about the existence of model structure, but to prove that the fibrations are precisely the Kan fibrations (and also to prove all the good properties of $Ex^\infty$ without using topological spaces); for two different proofs of this fact using $Ex^\infty$, see Prop. 2.1.41 as well as Scholium 2.3.21 for an alternative). For the rest, everything was already in the book of Gabriel and Zisman, for instance.
Finally, I would even add that, in Quillen's original paper, the model structure on topological spaces in obtained by transfer from the model structure on simplicial sets. And that is indeed a rather natural way to proceed.
Best Answer
Just for fun, here's a purely abstract way of seeing neither of these can happen:
The functors you wrote down both preserve weak equivalences, so are morphisms of relative categories, and induce adjunctions of the associated $\left(\infty,1\right)$-categories, which are $\infty$-groupoids and $1$-groupoids (which is actually a $(2,1)$-category) respectively. The adjunction is precisely the one exhibiting $1$-groupoids as a reflective subcategory of $\infty$-groupoids. The full and faithful inclusion is modeled by $N$. The $\left(\infty,1\right)$-category of $\infty$-groupoids is freely generated by the (any) contractible $\infty$-groupoid (i.e. the one point space), (i.e. it's the free colimit cocompletion of the terminal category). Since the $\left(2,1\right)$-category of groupoids is cocomplete, and contains the terminal groupoid, if the inclusion preserved colimits, it would have to be essentially surjective:
If $X$ is any $\infty$-groupoid, you can write $X$ as the colimit of the terminal functor $X \to \infty\mbox{-}\mathbf{Gpd}$ (the constant functor with value the point) (this is basically just the Yoneda Lemma for $\infty$-categories), and this functor (obviously) factors through the inclusion $\mathbf{Gpd} \hookrightarrow \infty\mbox{-}\mathbf{Gpd}$.
If the left adjoint $\tau_1$ of this inclusion (which is modeled by $\Pi_1$) preserved even finite limits, then this would exhibit the $(2,1)$-category of groupoids as a left-exact localization of $\infty\mbox{-}\mathbf{Gpd}=\mathbf{Psh}_\infty\left(*\right),$ i.e. it would exhibit $\mathbf{Gpd}$ as an $\infty$-topos. This means there would exist a unique Grothendieck topology $J$ on the terminal category such that $\mathbf{Gpd}$ sat somewhere between $\mathbf{Sh}_\infty\left(*,J\right)$ and its hypercompletion $\mathbf{HSh}_\infty\left(*,J\right).$ But, there is a unique Grothendieck topology on the terminal category (the trivial one!), so we'd have $$\mathbf{HSh}_\infty\left(*,J\right)=\mathbf{Sh}_\infty\left(*,J\right)=\mathbf{Psh}_\infty\left(*\right).$$ So we'd have to have $\mathbf{Gpd}\simeq \mathbf{Psh}_\infty\left(*\right)=\infty\mbox{-}\mathbf{Gpd}$ which is clearly nonsense.