Fix a multivariate polynomial $\mathfrak{p}$ in $n$ non-commuting variables of arbitrary degree, and consider $n$ independent $N\times N$ complex Ginibre matrices $X_1^N,\cdots,X_n^N$. We prove that the empirical spectral distribution of $P^N=\mathfrak{p}(X_1^N,\cdots,X_n^N)$ converges as $N$ tends to infinity to the so-called Brown measure of $\mathfrak{p}$ evaluated at free circular variables. For polynomials of degree at most 2, the convergence was proven by Cook, Guionnet, and Husson \cite{cook2022spectrum}, and we prove that the convergence in fact holds for polynomials $\mathfrak{p}$ of any degree. The main step in the proof is a least singular value lower bound for $P^N-z$ for almost all complex shifts $z$, and we prove this via a least singular value lower bound for a wide class of tensorized Ginibre matrices of finite type with a deterministic shift, which is of independent interest. We further show that the Brown measure convergence holds beyond Gaussians: the same convergence holds when the entry law has mean 0, variance 1, bounded density on $\mathbb{C}$ and finite moments of all orders.