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Detection of primordial gravitational-wave backgrounds generated during the early universe phase transitions is a key science goal for future ground-based detectors. The rate of compact binary mergers is so large that their cosmological population produces a confusion background that could masquerade the detection of potential primordial stochastic backgrounds. In this paper we study the ability of current and future detectors to resolve the confusion background to reveal interesting primordial backgrounds. The current detector network of LIGO and Virgo and the upcoming KAGRA and LIGO-India will not be able to resolve the cosmological compact binary source population and its sensitivity to stochastic background will be limited by the confusion background of these sources. We find that a network of three (and five) third generation (3G) detectors of Cosmic Explorer and Einstein Telescope will resolve the confusion background produced by binary black holes leaving only about 0.013\% (respectively, 0.00075\%) unresolved; in contrast, as many as 25\% (respectively, 7.7\%) of binary neutron star sources remain unresolved. Consequently, the binary black hole population will likely not limit observation of primordial backgrounds but the binary neutron star population will limit the sensitivity of 3G detectors to $Ω_{\rm GW} \sim 10^{-11}$ at 10 Hz (respectively, $Ω_{\rm GW} \sim 3\times 10^{-12}$).