Context. Over recent years, additional low-amplitude non-radial modes have been detected in many first-overtone RR Lyrae stars. These non-radial modes form a characteristic period ratio with the dominant first overtone of around 0.61. The incidence rate of this phenomenon varies from population to population. It is also strongly dependent on the quality of the analyzed data. Current models aimed at explaining these additional signals involve non-radial modes of degrees of 8 and 9. Aims. Using synthetic horizontal branch populations, we investigate the incidence rate of first-overtone RR Lyrae stars with non-radial modes, depending on the population properties, namely, ages and metallicities. We compare our results with the observed results for globular clusters and the vast collection of field first-overtone RR Lyrae stars to test the model predictions. Methods. We used synthetic horizontal branches combined with pulsation models to predict how the incidence rate would depend on the age and metallicity of the population. To test whether the results based on synthetic horizontal branches are realistic, we compared them to incidence rates observed by TESS in first-overtone field RR Lyrae stars, using photometric metallicity values from a newly established calibration for TESS. Results. The analysis of synthetic horizontal branches indicates that the incidence rate decreases with decreasing metallicity. We inferred the photometric metallicity for RR Lyrae stars observed by TESS and showed that the theoretical predictions are in agreement with the observations. Using the same method, we also concluded that the metallicity distribution of RR Lyrae stars showing an additional mode with a period-ratio around 0.68 appears to be different from that of 1) all first-overtone stars and 2) those showing additional non-radial modes.