The fate of the Fermi surface in bulk electron-doped Sr$_{2}$IrO$_{4}$ remains elusive, as does the origin and extension of its pseudogap phase. Here, we use high-resolution angle-resolved photoelectron spectroscopy (ARPES) to investigate the electronic structure of Sr$_{2-x}$La$_{x}$IrO$_{4}$ up to $x=0.2$, a factor of two higher than in previous work. Our findings reveal that the Fermi surface evolves smoothly with doping. Notably, the antinodal pseudogap persists up to the highest doping level, while nodal quasiparticle coherence increases monotonously. This demonstrates that the sharp increase in Hall carrier density recently observed above $x^{*}=0.16$ [Y.-T. Hsu et al., Nature Physics 20, 1596 (2024)] cannot be attributed to the closure of the pseudogap. Further, we determine a temperature boundary of the pseudogap of $T^{*}\simeq~200~\textrm{K}$ for $x=0.2$, comparable to cuprates. Our results suggest that pseudogaps are a generic feature of doped quasi-2D antiferromagnetic Mott insulators, likely related to short range magnetic correlations.