We forecast constraints on the amplitude of matter clustering $\sigma_8(z)$ achievable with the combination of cluster weak lensing and number counts, in current and next-generation weak lensing surveys. We advocate an approach, analogous to galaxy–galaxy lensing, in which the observables in each redshift bin are the mean number counts and the mean weak lensing profile of clusters above a threshold in an observable proxy of halo mass. The primary astrophysical nuisance parameter is the logarithmic scatter $\sigma_\mathrm{ln M}$ between the mass proxy and true halo mass near the threshold. For surveys with parameters similar to those of the Dark Energy Survey (DES), the Roman Space Telescope High Latitude Survey (HLS), and the Rubin Observatory Legacy Survey of Space and Time (LSST), we forecast aggregate precision on $\sigma_8$ of 0.26%, 0.24%, and 0.10%, respectively, if the mass–observable scatter is known externally to $\Delta\sigma_\mathrm{ln M} \leq 0.01$. These constraints would be degraded by about 20% for $\Delta\sigma_\mathrm{ln M} = 0.05$ in the case of DES or HLS and for $\Delta\sigma_\mathrm{ln M} = 0.016$ for LSST. A one-month observing program with Roman Space Telescope targeting $\sim 2500$ massive clusters could achieve a $\sim 0.5%$ constraint on $\sigma_8(z=0.7)$ on its own, or a $\sim 0.33%$ constraint in combination with the HLS. Cluster weak lensing is potentially one of the most powerful probes of dark energy, modified gravity, or decaying dark matter models. Realizing this potential requires accurate knowledge of the relation between cluster observables and halo mass and stringent control of observational and theoretical systematics. We provide analytic approximations to our numerical results that allow easy scaling to other survey assumptions or other methods of cluster mass estimation.