Please help me resolve the following paradox:

False claim: Let $X$ be an $n$-coskeletal, $n$-connected simplicial set. Then $X$ is weakly contractible.

Actually, I suppose the claim is true when $n=0$; but it is false for $n\geq 1$: nerves of posets realize all homotopy types, and these are 1-coskeletal.

Paradoxical "Proof" of false claim: Fix $m > n$. To show that $\pi_m(X) = 0$, consider a map $f: Sd^k \partial \Delta^{m+1} \to X$; we want to show that for some $l$, $Sd^{k+l} \partial \Delta^{m+1} \to Sd^{k} \partial \Delta^{m+1} \xrightarrow f X$ extends along $Sd^{k+l} \partial \Delta^{m+1} \to Sd^{k+l} \Delta^{m+1}$. Here $Sd$ is barycentric subdivision. Since $X$ is $n$-coskeletal, it suffices to show this after passing to $n$-skeleta. But then this problem can be solved because $X$ is $n$-connected, so that (after subdivision) lifts exist along any map between $n$-skeletal simplicial sets.

Resolution: The problem with the "proof" is that subdivision fails to commute with taking skeleta.

Paradoxical salvaged "proof": One can still try to build an extension of $f$ as follows. First, find an extension along $Sd^{l_1} sk_n Sd^k \partial \Delta^{m+1} \to Sd^{l_1} sk_n Sd^k \Delta^{m+1}$. To turn this into an extension along $sk_n Sd^{k+l_1} \partial \Delta^{m+1} \to sk_n Sd^{k+l_1} \Delta^{m+1}$ involves another lifting problem along a map between $n$-skeletal simplicial sets, so it can be solved after applying $Sd^{l_2}$. Continue in this manner, and you eventually construct an extension of $f$ along $Sd^\infty \partial \Delta^{m+1} \to Sd^\infty \Delta^{m+1}$. This shows that the homotopy group represented by $f$ is trivial.

Attempted resolution: The process doesn't converge at a finite stage. The loophole must be that $Sd^\infty \partial \Delta^{m+1}$ doesn't have the homotopy type of $\partial \Delta^{m+1}$.

Questions:

1. Is the map $\partial \Delta^{m+1} \to Sd^\infty \partial \Delta^{m+1} := \varinjlim_l Sd^l \partial \Delta^{m+1}$ a weak homotopy equivalence?

2. If so, then how does one actually resolve the revived paradox?

3. Even if not, is there a better way to formulate the resolution of the revived paradox? It feels as though a loophole was exploited.

4. I think I'm starting to see why by the naure of "subdivision", no such operator is going to commute with taking skeleta. But if someone has a nice way to formulate why this is so, I'd love to hear it.