The reference in NYT appears to be to the authors of the paper Carson, Miller and Sassler 2018. Among other things, this paper gives analysis of the statistical relationship between division of household work and relationship trouble. I have been unable to find the descriptive data for the interface of these two variables in the paper (and there are not supplementary materials given), so I suspect that the data used in the graph may have been obtained by the journalist inquiring directly with the authors.
Since there is nothing in the graph to stipulate that these are conditional percentages, it appears more likely that these are raw percentages referring to each joint-category (i.e., percentages of all women participants in the survey that fall into the relevant column and row jointly). It is important to note that the three categories of marital discord are not mutually exclusive (e.g., a relationship that has "become physical" is probably also reported as being "in trouble"), so this is why the percentages add up to substantially more than 100%.
Assuming that this interpretation is correct, the main deficiency of the graph is that it does not show the full data on the relevant variables, since it excludes the category of cases where the man does most of the dishes. The graph could be improved by adding a final column of percentages for these cases. Without this column, it is not easy to see whether the statistical relationship between the variables is consistent across all three categories (of dishwashing work) or not. If one were looking for greater completeness, the graph could also be supplemented by augmenting it with a second graph showing the outcomes reported by the men. Since this descriptive data is not made available in the paper, I cannot generate this chart, but it would be possible to do so if you contact the authors of the paper to obtain the required statistics.
If desired, the graph could be visually simplified by presenting it as a "bubble plot", which would present the size of the percentages as bubbles (with area proportional to the percentage), rather than presenting them as pie-wheels. This is really a matter of aesthetic preference --- most statisticians prefer to use the simplest possible visualisation, but NYT probably has its own "house style" for these kinds of graphs.
Finally, it is worth noting that this kind of graph only presents a univariate statistical relationship between a pair of variables. The paper from which this is taken actually conducts a multivariate analysis, which conditions on a range of relationship variables. In Table 4, the authors present estimated logistic-regression coefficients estimating the conditional association between dishwashing labour and relationship troubles. These coefficient estimates give a better idea of the estimated relationship between the variables, since they condition on other relationship variables. Arguably, any kind of graph of the pairwise relationships in the descriptive data has the potential to be a bit misleading, insofar as it fails to condition on all the variables in the analysis. (This is complicated by the fact that average newspaper readers have difficulty reading and understanding graphs of coefficient estimates from regression analysis; presumably the NYT has decided that a simpler graph showing univariate association is preferable to a more complicated graph showning conditional association.)