Just wanted to say here that in English, "money" is always singular while in Greek, both singular and plural forms exist. It's just that plural is used in the most cases. Το χρήμα is singular and τα χρήματα plural. That means that you can also say "το χρήμα δεν φέρνει την ευτυχία" and the meaning will be the same: "money doesn't bring/buy happiness". The same happens with the noun "hair". In English it's always singular but in Greek, it can be either singular (το μαλλί) or plural (τα μαλλιά). Again, plural is more common. For example, if you want to say "my hair is brown", both "το μαλλί μου είναι καφέ" and "τα μαλλιά μου είναι καφέ" are correct.
Here's a fun fact (that I actually didn't notice until today.)
Joy and happiness are considered to be very different terms in English.
Joy (a direct translation to χαρά) is considered to be much stronger and more internal. It's usuallly triggered by other emotions, such as thankfullness or caring for other people, or even other people's emotions, such as other people's happiness or joy.
Happiness (a direct translation to ευτυχία) is considered to be more temporary. It's usually triggered by 'external' factors, such as material objects, things, people, places, events, experiences.
The difference between these words in the same as it is with χαρά and ευτυχία, but we do consider ευτυχία to be a stronger feeling than χαρά, so you could say the meanings are opposite.
So, "Τα χρήματα δεν φέρνουν την ευτυχία" would make perfect sense in Greek, since money can keep you 'externally' satisfied, but not always 'internally', but "Money does not bring hapiness" would make much more sense if it was "Money does not bring joy" instead, since joy is considered to be the stronger term.
So, I do believe joy, while being a direct translation to χαρά, and despite the fact that this sentence is technically a fixed one, should be accepted as a translation, for context's sake. :)