The plot shown in Why is the white-dwarf mass distribution highly peaked? is for single white dwarfs found in the SDSS survey.
Single white dwarfs are not the progenitors of type Ia supernovae. Single star evolution produces white dwarfs that are limited to $\leq 1.2 M_\odot$ or thereabouts, which is significantly below the likely mass threshold for producing a supernovae ($>1.35M_\odot$).
The progenitors of type Ia supernovae are most likely those that exist in binary systems, where the white dwarf can accrete mass from a companion star and hence approach the threshold for ignition of a supernova. The white dwarfs in such systems do not feature in the plot in the question you refer to.
Indeed, when you look at white dwarfs in close, interacting binary systems, their mass distribution is displaced towards higher masses (e.g. Wijnen 2015), however, the growth in mass that is seen is actually quite difficult to reconcile with models of how these short period binary systems have evolved. There are also estimates of the masses of the white dwarfs in some of these systems (classical novae objects) that are very close to the critical mass (e.g. $1.35-1.37M_\odot$, Hachisu & Kato 2001). There are also now observations of rare, single white dwarfs with inferred masses very close to the Chandrasekhar mass, which by virtue of their fast rotation and strong magnetic fields are thought to be a merger product of two white dwarfs of lower mass (e.g. Caiazzo et al. 2021; Kilic et al. 2021).
