Two decades of work have shown that the physics of Tb-based pyrochlores is controlled by a subtle equilibrium between quadrupole-quadrupole and dipolar-dipolar magnetic interactions, as exemplified by the ordered spin ice Tb₂Sn₂O₇ and the quantum spin liquid candidate Tb₂Ti₂O₇. The high-entropy approach is thus quite promising, as it offers the possibility of a delicate tuning of chemical disorder. In this work, we investigate the entropy-stabilized pyrochlore compound Tb₂(TiZrHfGeSn)₂O₇. We report the lack of long range magnetic order, yet the observed magnetic diffuse scattering is characteristic of antiferromagnetic first-neighbor correlations. The crystal field excitation spectrum, with broaden levels, visibly reflects the smooth environmental disorder of the Tb environment. The low energy dynamics are characterized by a narrow mode at about 0.4 meV, consistent with specific heat. Remarkably, as illustrated by a model of random isotropic shifts of oxygen atoms around Tb ions, the spectral weight of this mode is a direct consequence of deviations from the D_3d symmetry at Tb sites. In the light of these results, quadrupolar interactions are also discussed.