We consider a cross-diffusion model of tumor growth structured by phenotypic trait. We prove the existence of weak solutions and the incompressible limit as the pressure becomes stiff extending methods recently introduced in the context of two-species cross-diffusion systems. Moreover, we recover additional regularity estimates. We show that an L2-version of the celebrated Aronson-Bénilan estimate extends to structured models. As a consequence, we recover a sharp L1-bound on the Laplacian of the pressure. In particular, we are able to remove a technical constraint on the reaction terms assumed by Gwiazda et al. for the two-species model, by proving a new L4-bound on the pressure gradient.