We present estimates for the number of shadow-resolved supermassive black hole (SMBH) systems that can be detected using radio interferometers, as a function of angular resolution, flux density sensitivity, and observing frequency. Accounting for the distribution of SMBHs across mass, redshift, and accretion rate, we use a new semianalytic spectral energy distribution model to derive the number of SMBHs with detectable and optically thin horizon-scale emission. We demonstrate that (sub)millimeter interferometric observations with similar to 0.1 mu as resolution and similar to 1 mu Jy sensitivity could access >10(6) SMBH shadows. We then further decompose the shadow source counts into the number of black holes for which we could expect to observe the first- and second-order lensed photon rings. Accessing the bulk population of first-order photon rings requires less than or similar to 2 mu as resolution and less than or similar to 0.5 mJy sensitivity, whereas doing the same for second-order photon rings requires less than or similar to 0.1 mu as resolution and less than or similar to 5 mu Jy sensitivity. Our model predicts that with modest improvements to sensitivity, as many as similar to 5 additional horizon-resolved sources should become accessible to the current Event Horizon Telescope (EHT), whereas a next-generation EHT observing at 345 GHz should have access to similar to 3 times as many sources. More generally, our results can help guide enhancements of current arrays and specifications for future interferometric experiments that aim to spatially resolve a large population of SMBH shadows or higher-order photon rings.