Though most of the paper's studies are based on observational and survey data, which is typically where the 'correlation is not causation' criticism is most effective, the authors actually perform a experiment. They give $40 to two groups of participants and task them to make a time-saving purchase and a material purchase, and survey their happiness after each purchase.
Their results from this experiment are inconsistent with your proposed hypothesis as they've controlled for baseline happiness by randomizing the groups. See materials and methods, study 8 from the linked article.
Edit: it's also in the abstract! "A field experiment provides causal evidence that working adults report greater happiness after spending money on a time-saving purchase than on a material purchase."
"In contrast, for respondents who spent money on time-saving purchases (n = 804), the negative effect of time pressure on life satisfaction was relatively weak, B = −0.03, Z = 1.46, P = 0.144, 95% CI (−0.08, 0.01). These results suggest that using money to buy time indeed buffers people from the negative effects of time stress on well-being."