Context. The intracluster medium (ICM) is the low-density diffuse gas that fills the space between galaxies within galaxy clusters. It is primarily composed of magnetized plasma, which reaches virial temperatures of up to 10(8) K, probably due to mergers of subhalos. Under these conditions, the plasma is weakly collisional and therefore has an anisotropic pressure tensor with respect to the local direction of the magnetic field. This triggers very fast, Larmor-scale, pressure-anisotropy-driven kinetic instabilities that alter magnetic field amplification. Aims. We aim to study magnetic field amplification through a turbulent, small-scale dynamo, including the effects of the kinetic instabilities, during the evolution of a typical massive galaxy cluster. A specific aim of this work is to establish a redshift limit from which a dynamo has to start to amplify the magnetic field up to equipartition with the turbulent velocity field at redshift z = 0. Methods. We implemented one-dimensional radial profiles for various plasma quantities for merger trees generated with the modified GALFORM algorithm. We assumed that turbulence is driven by successive mergers of dark matter halos and constructed effective models for the Reynolds number Re-eff dependence on the magnetic field in three different magnetization regimes (unmagnetized, magnetized "kinetic", and magnetized "fluid"), including the effects of kinetic instabilities. The magnetic field growth rate is calculated for the different Reeff models. Results. The model results in a higher magnetic field growth rate at higher redshift. For all scenarios considered in this study, to reach equipartition at z = 0, it is sufficient for the amplification of the magnetic field to start at redshift z(start) approximate to 1.5 and above. The time to reach equipartition can be significantly shorter in cases with systematically smaller turbulent forcing scales and for the highest Re-eff models. Conclusions. The origin of magnetic fields in the weakly collisional ICM can be explained by the small-scale turbulent dynamo, provided that the dynamo process starts beyond a given redshift. Merger trees are useful tools for studying the evolution of magnetic fields in weakly collisional plasmas, and could also be used to constrain the different stages of the dynamo that could potentially be observed by future radio telescopes.