  The transition metal dichalcogenide $1T$-TiSe$_2$ is a quasi-two-dimensional
layered material with a phase transition towards a commensurate charge density
wave (CDW) at a critical temperature T$_{c}\approx 200$K. The relationship
between the origin of the CDW instability and the semimetallic or
semiconducting character of the normal state, i.e., with the non-reconstructed
Fermi surface topology, remains elusive. By combining angle-resolved
photoemission spectroscopy (ARPES), scanning tunneling microscopy (STM), and
density functional theory (DFT) calculations, we investigate
$1T$-TiSe$_{2-x}$S$_x$ single crystals. Using STM, we first show that the
long-range phase coherent CDW state is stable against S substitutions with
concentrations at least up to $x=0.34$. The ARPES measurements then reveal a
slow but continuous decrease of the overlap between the electron and hole
($e$-$h$) bands of the semimetallic normal-state well reproduced by DFT and
related to slight reductions of both the CDW order parameter and $T_c$. Our DFT
calculations further predict a semimetal-to-semiconductor transition of the
normal state at a higher critical S concentration of $x_c$=0.9 $\pm$0.1, that
coincides with a melted CDW state in TiSeS as measured with STM. Finally, we
rationalize the $x$-dependence of the $e$-$h$ band overlap in terms of
isovalent substitution-induced competing chemical pressure and charge
localization effects. Our study highlights the key role of the $e$-$h$ band
overlap for the CDW instability.
