The collection of planetary system properties derived from large surveys such as Kepler provides critical constraints on planet formation and evolution. These constraints can only be applied to planet formation models, however, if the observational biases and selection effects are properly accounted for. Here we show how epos, the Exoplanet Population Observation Simulator, can be used to constrain planet formation models by comparing the Bern planet population synthesis models to the Kepler exoplanetary systems. We compile a series of diagnostics, based on occurrence rates of different classes of planets and the architectures of multiplanet systems within 1 au, that can be used as benchmarks for future and current modeling efforts. Overall, we find that a model with 100-seed planetary cores per protoplanetary disk provides a reasonable match to most diagnostics. Based on these diagnostics we identify physical properties and processes that would result in the Bern model more closely matching the known planetary systems. These are as follows: moving the planet trap at the inner disk edge outward; increasing the formation efficiency of mini-Neptunes; and reducing the fraction of stars that form observable planets. We conclude with an outlook on the composition of planets in the habitable zone, and highlight that the majority of simulated planets smaller than 1.7 Earth radii in this zone are predicted to have substantial hydrogen atmospheres. The software used in this paper is available online for public scrutiny at https://github.com/GijsMulders/epos.