I've been looking at Kunen's (2011, p. 227) proof that the Diamond Principle plus:
$(\Diamond^+)$ There is a sequence $\mathcal A = \langle A_\alpha:\alpha<\omega_1\rangle$ such that $A_\alpha$ is a countable subset of $\mathcal P(\alpha)$ and for every $A\subseteq \omega_1$ there is a club $C\subseteq\omega_1$ such that for all $\alpha\in C$, $A\cap \alpha, C\cap\alpha\in A_\alpha$.
implies that there is a Kurepa family, i.e.:
an $\mathcal F\subseteq \mathcal P(\omega_1)$ such that $|\mathcal F|>\omega_1$ and for all $\alpha<\omega_1$ $|\mathcal F\upharpoonleft\alpha|\leq\omega$ (where $\mathcal F\upharpoonleft\alpha = \{X\cap\alpha:X\in\mathcal F\}$).
It seems to me that there's a simpler proof. Let $\mathcal F$ be the set of all club sets witnessing $\Diamond^+$. Either $|\mathcal F|>\omega_1$ or not.
(1) $|\mathcal F|>\omega_1$. Then I claim that $\mathcal F$ is a Kurepa family. For suppose that $|\mathcal F\upharpoonleft \alpha|>\omega$ for some $\alpha<\omega_1$. WLOG, let $\alpha$ be the least such. Since each $X\in\mathcal F$ is club, if $X\cap \alpha$ is unbounded, then $\alpha\in X$ and $X\cap \alpha\in A_\alpha$. So there are at most countably many $X\cap \alpha$ unbounded in $\alpha$. For the rest, they contain a greatest ordinal $\beta<\alpha$. By the pigeon hole principle, uncountably many have the same greatest ordinal $\beta<\alpha$. But then $\mathcal F\upharpoonleft \beta$ is uncountable, contradicting the minimality of $\alpha$.
(2) $|\mathcal F|\leq\omega_1$. Then, by the pigeon hole principle, there is some $C\in\mathcal F$ for which there are at least $\omega_2$ many subsets of $\omega_1$ $A$ for which $A\cap \alpha\in \mathcal A_\alpha$ for $\alpha\in C$. So, those subsets are a Kurepa family.
Am I missing something?