The Cherenkov Telescope Array (CTA) is about to enter construction phase and one of its main key science projects is to perform an unbiased survey in search of extragalactic sources. We make use of both the latest blazar gamma-ray luminosity function and spectral energy distribution to derive the expected number of detectable sources for both the planned Northern and Southern arrays of the CTA observatory. We find that a shallow, wide survey of about 0.5 hour per field of view would lead to the highest number of blazar detections. Furthermore, we investigate the effect of axion-like particles and secondary gamma rays from propagating cosmic rays on the source count distribution, since these processes predict different spectral shape from standard extragalactic background light attenuation. We can generally expect more distant objects in the secondary gamma-ray scenario, while axion-like particles do not significantly alter the expected distribution. Yet, we find that, these results strongly depend on the assumed magnetic field strength during the propagation. We also provide source count predictions for the High Altitude Water Cherenkov observatory (HAWC), the Large High Altitude Air Shower Observatory (LHAASO) and a novel proposal of a hybrid detector.