The thermal Sunyaev-Zel'dovich (tSZ) effect is a powerful probe of cosmology. The statistical errors in the tSZ power spectrum measurements are dominated by the presence of massive clusters in a survey volume that are easy to identify on an individual cluster basis. First, we study the impact of super sample covariance (SSC) on the tSZ power spectrum measurements, and find that the sample variance is dominated by the connected non-Gaussian (cNG) covariance arising mainly from Poisson number fluctuations of massive clusters in the survey volume. Second, we find that removing such individually detected, massive clusters from the analysis significantly reduces the cNG contribution, thereby causing the SSC to be a leading source of the sample variance. We then show, based on a Fisher analysis, that the power spectrum measured from the remaining diffuse tSZ effects can be used to obtain tight constraints on cosmological parameters as well as the hydrostatic mass bias parameter. Our method allows the use of both individual tSZ cluster counts and the power spectrum measurements of diffuse tSZ signals for cosmology and intracluster gas physics.