We study the convergence of the empirical spectral distribution of $\mathbf{A} \mathbf{B} \mathbf{A}$ for $N \times N$ orthogonal projection matrices $\mathbf{A}$ and $\mathbf{B}$, where $\frac{1}{N}\mathrm{Tr}(\mathbf{A})$ and $\frac{1}{N}\mathrm{Tr}(\mathbf{B})$ converge as $N \to \infty$, to Wachter's MANOVA law. Using free probability, we show mild sufficient conditions for convergence in moments and in probability, and use this to prove a conjecture of Haikin, Zamir, and Gavish (2017) on random subsets of unit-norm tight frames. This result generalizes previous ones of Farrell (2011) and Magsino, Mixon, and Parshall (2021). We also derive an explicit recursion for the difference between the empirical moments $\frac{1}{N}\mathrm{Tr}((\mathbf{A} \mathbf{B} \mathbf{A})^k)$ and the limiting MANOVA moments, and use this to prove a sufficient condition for convergence in probability of the largest eigenvalue of $\mathbf{A} \mathbf{B} \mathbf{A}$ to the right edge of the support of the limiting law in the special case where that law belongs to the Kesten-McKay family. As an application, we give a new proof of convergence in probability of the largest eigenvalue when $\mathbf{B}$ is unitarily invariant; equivalently, this determines the limiting operator norm of a rectangular submatrix of size $\frac{1}{2}N \times αN$ of a Haar-distributed $N \times N$ unitary matrix for any $α\in (0, 1)$. Unlike previous proofs, we use only moment calculations and non-asymptotic bounds on the unitary Weingarten function, which we believe should pave the way to analyzing the largest eigenvalue for products of random projections having other distributions.