Having implemented the option to make ad-hoc modifications of cross-section, multiplicity, and elasticity of hadronic interactions during the development of cosmic-ray showers into the CORSIKA simulation package in a consistent way for both primary protons and arbitrary nuclei, we can study the impact of these modifications on a variety of observables for different primaries. The modifications of the interactions are generic, without any reference to a specific mechanism. The ranges of parameters are only restricted by existing experimental data from accelerators and nuclear emulsions. The complete 3-dimensional view of the showers provided by CORSIKA allows us to study not only the depth of the maximum of the longitudinal development of the showers and total particle yield at the ground but also to see how the modifications impact the signal at different distances from the shower core and at different energy thresholds for the detection of secondary particles at the ground. Furthermore, we investigate the changes in the maximum depth where muons are produced, the expected rates of anomalous shower profiles, the correlation between the maximum of the longitudinal profiles and ground signal, and the fluctuations of all the observables. We show not only which modifications help the most to alleviate the tension between the current hadronic interaction models and data from ultra-high energy cosmic ray observatories, most prominently the Pierre Auger Observatory, but also which observables are the most sensitive to the individual parameters of the hadronic interactions, thus helping to guide the design of future cosmic-ray observatories.