In the Bi cuprates, the presence of a near 1$times$5 superstructure is well known. Usually, this superstructure is suppressed by the substitution of lead, but there have been reports of a phase separation in so called {alpha} and {beta} phases. This paper shows in high detail time how and why the phase separation develops and what happens to the quasi-1$times$5 superstructure upon lead substitution. For this purpose, the lanthanum- and lead-substituted single-layered superconductor Bi$_{2+z}$Sr$_{2-z}$CuO$_{6+delta}$ has been investigated by scanning tunneling microscopy and low-energy electron diffraction. The La content was kept constant at slightly under-doped concentration while the Pb content was changed systematically. Thermodynamic considerations show that a phase mixture of {alpha} and {beta} phases is inevitable.