Here are two suggestions, whether they follow your instructions of "no machinery allowed", I will leave to you.

The following seems fairly direct to me, and I would think is the "standard" answer to your question:

First, partially order $\mathbb N^{\mathbb N}$ by *eventual domination*: $f<g$ iff $f(n)<g(n)$ for all $n$ large enough.

If $X\subseteq\mathbb N^{\mathbb N}$ is unbounded and countably directed, then there are $f\ne g$ in $X$ such that $f(n)\le g(n)$ for *all* $n$. This is proved as Lemma 0.7 in Todorcevic's **Partition problems in topology**.

From this it follows that under $\mathsf{OCA}$, any subset of $\mathbb N^{\mathbb N}$ of size $\aleph_1$ is bounded. (For example, this follows from Proposition 8.4.(c) and the proof of Theorem 3.5 in Todorcevic's book.)

(Refining this argument (eventually) gives us that $\mathsf{OCA}$ implies that ${\mathfrak b}=\aleph_2$.) The only technical tool this uses is the notion of *oscillation*, and what is needed can be developed directly in the proof of the statements above.

A different approach goes by building on Proposition 8.4.(c) directly: $\mathsf{OCA}$ implies that if $X,Y$ are uncountable sets of reals, then there is an uncountable subset of $X$ and a strictly increasing map of this subset into $Y$. This easily clashes with $\mathsf{CH}$, for example, using the results of Sierpiński on *Sur un problème concernant les types de dimensions*.