I have my own little saying that I always say in response to that saying: Lack of money can definitely cause unhappiness though

You don't need to look further than Maslow's hierarchy of needs to see why, when you don't have enough money to reliably fulfill some of the most fundamental needs like shelter and food, it's going to severely hinder your self actualization. Forget about moving on to the more psychological higher-level needs when you are worrying about where you are going to sleep tonight. It's easy to understand why happiness might be very hard to reach for individuals who don't have enough money to cover their basic needs