The dust-to-stellar mass ratio ($M_{rm dust}$/$M_{rm star}$) is a crucial yet poorly constrained quantity to understand the production mechanisms of dust, metals and stars in galaxy evolution. In this work we explore and interpret the nature of $M_{rm dust}$/$M_{rm star}$ in 300 massive ($M_{star}>10^{10}M_{odot}$), dusty star-forming galaxies detected with ALMA up to $zapprox5$. We find that $M_{rm dust}$/$M_{rm star}$ evolves with redshift, stellar mass, specific SFR and integrated dust size, differently for main sequence and starburst galaxies. In both galaxy populations $M_{rm dust}$/$M_{rm star}$ rises until $zsim2$ followed by a roughly flat trend towards higher redshifts. We show that the inverse relation between $M_{rm dust}$/$M_{rm star}$ and $M_{star}$ holds up to $zapprox5$ and can be interpreted as an evolutionary transition from early to late starburst phases. We demonstrate that $M_{rm dust}$/$M_{rm star}$ in starbursts mirrors the increase in molecular gas fraction with redshift, and is enhanced in objects with the most compact dusty star-formation. The state-of-the-art cosmological simulation SIMBA broadly matches the evolution of $M_{rm dust}$/$M_{rm star}$ in main sequence galaxies, but underestimates it in starbursts. The latter is found to be linked to lower gas-phase metallicities and longer dust growth timescales relative to data. Our data are well reproduced by analytical model that includes recipes for rapid metal enrichment, strongly suggesting that high $M_{rm dust}$/$M_{rm star}$ is due to fast grain growth in metal enriched ISM. Our work highlights multifold benefits of using $M_{rm dust}$/$M_{rm star}$ as a diagnostic tool for: (1) separating main sequence and starburst galaxies until $zsim5$; (2) probing the evolutionary phases of dusty galaxies, and (3) refining the treatment of dust life cycle in simulations.
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