The discovery of fast moving stars in the Milky Way's most massive globularcluster, $\omega$Centauri ($\omega$Cen), has provided strong evidence for anintermediate-mass black hole (IMBH) inside of it. However, $\omega$Cen is knownto be the stripped nuclear star cluster (NSC) of an ancient, now-destroyed,dwarf galaxy. The best candidate to be the original host progenitor of$\omega$Cen is the tidally disrupted dwarf $Gaia$-Sausage/Enceladus (GSE), aformer Milky Way satellite as massive as the Large Magellanic Cloud. I compare$\omega$Cen/GSE with other central BH hosts and place it within the broadercontext of BH-galaxy (co)evolution. The IMBH of $\omega$Cen/GSE follows thescaling relation between central BH mass and host stellar mass (${\rm M}_{\rmBH}{-}{\rm M}_\star$) extrapolated from local massive galaxies (${\rm M}_\star\gtrsim 10^{10}\,{\rm M}_\odot$). Therefore, the IMBH of $\omega$Cen/GSEsuggests that this relation extends to the dwarf-galaxy regime. I verify that$\omega$Cen (GSE), as well as other NSCs with candidate IMBHs and ultracompactdwarf galaxies, also follow the ${\rm M}_{\rm BH}{-}\sigma_\star$ relation withstellar velocity dispersion. Under the assumption of a direct collapse BH,$\omega$Cen/GSE's IMBH would require a low initial mass ($\lesssim$10,000 ${\rmM}_{\odot}$) and almost no accretion over $\sim$3 Gyr, which could be theextreme opposite of high-$z$ galaxies with overmassive BHs such as GN-z11. If$\omega$Cen/GSE's IMBH formed from a Population III supernova remnant, then itcould indicate that both light and heavy seeding mechanisms of central BHformation are at play. Other stripped NSCs and dwarf galaxies could helpfurther populate the ${\rm M}_{\rm BH}{-}{\rm M}_{\star}$ and ${\rm M}_{\rmBH}{-}\sigma_\star$ relations in the low-mass regime and constraint IMBHdemographics and their formation channels.