Further development of quantum emitter based communication and sensing applications intrinsically depends on the availability of robust single-photon detectors. Here, we demonstrate a new generation of superconducting single-photon detectors specifically optimized for the 500-1100 nm wavelength range, which overlaps with the emission spectrum of many interesting solid-state atom-like systems, such as nitrogen-vacancy and silicon-vacancy centers in diamond. The fabricated detectors have a wide dynamic range (up to 350 million counts per second), low dark count rate (down to 0.1 counts per second), excellent jitter (62 ps), and the possibility of on-chip integration with a quantum emitter. In addition to performance characterization, we tested the detectors in real experimental conditions involving nanodiamond nitrogen-vacancy emitters enhanced by a hyperbolic metamaterial.