In short: Money and happiness are linked more subtly than the saying suggests: for most people happiness keeps rising with income, but money mainly removes the unhappiness of scarcity, not problems with other roots.
Simply put: For a long time it was claimed that above a certain income (about $75,000 a year in the US) money no longer increases happiness. A joint 2023 study by researchers who had disagreed (Killingsworth, Kahneman and Mellers) shows that for most people happiness keeps rising above that level; a "ceiling" applies to the unhappiest 20% or so. A Swedish study of lottery winners shows that a large win raises life satisfaction for the long term.
Example: A family with no reserve experiences every car breakdown as a crisis. Once it has saved €3,000, the next €600 repair is a nuisance, not a disaster — the money has not bought joy, but it has removed fear.
Watch out: ⚠ disputed. The 2023 study is large but based on the US; Bennedsen (2024, preprint) argues that the data show a plateau at around $200,000 a year. The lesson is not "money buys happiness" but: too little money brings great unhappiness; a lot of money does not necessarily bring great happiness.
Source: Killingsworth, Kahneman & Mellers, PNAS, 2023 — https://behavioralpolicy.princeton.edu/news/DK_wellbeing0323 · Bennedsen, arXiv:2401.05347, 2024 — https://arxiv.org/abs/2401.05347v1 · Lindqvist, Östling & Cesarini, 2020 — https://ideas.repec.org/a/oup/restud/v87y2020i6p2703-2726..html