The following is a folklore result.

Suppose $P$ is a countable support iteration of nontrivial forcings, $\langle P_\alpha, \dot{Q}_\alpha : \alpha < \omega_1 \rangle$. Then there is a complete embedding of $\mathrm{Add}(\omega_1)$ into $P$.

The forcing to add a Cohen subset of $\omega_1$ fails the $\omega_1$-approximation property, since it produces a “fresh” sequence— a sequence such that all initial segments are in the ground model.

In the 1979 paper, “Iterated perfect-set forcing,” Baumgartner and Laver seem to make a contrary claim. Lemma 6.2 states that the countable support iteration of Sacks forcing produces no fresh sequences of length some ordinal of uncountable cofinality. This is key to their argument that iterating Sacks forcing up to a weakly compact forces the tree property at $\omega_2$.

I do not see a flaw in their argument. Is the folklore claim correct? How is this resolved?