  Pressure evolution of the superconducting kagome metal CsV$_3$Sb$_5$ is
studied with single-crystal x-ray diffraction and density-functional
band-structure calculations. A highly anisotropic compression observed up to 5
GPa is ascribed to the fast shrinkage of the Cs-Sb distances and suppression of
Cs rattling motion. This prevents Sb displacements required to stabilize the
three-dimensional charge-density-wave (CDW) order and elucidates the
disappearance of the CDW already at 2 GPa despite only minor changes in the
electronic structure of the normal state. At higher pressures, vanadium bands
still change only marginally, whereas antimony bands undergo a major
reconstruction caused by the gradual formation of the interlayer Sb-Sb bonds.
Our results exclude pressure tuning of vanadium kagome bands as the main
mechanism for the non-trivial evolution of superconductivity in real-world
kagome metals. Concurrently, we establish the central role of Sb atoms in the
stabilization of a three-dimensional CDW and Fermi surface reconstruction.
