Maybe our system of money is eradicated in favor of some bizarre future version of bitcoin? I'll call it fitcoin. When Picard says, we work to better ourselves, maybe every Federation citizen has a chip installed that tracks neural connections, physical health, etc.
Working out, learning new languages and advanced science, performing tasks, etc. is loaded into a chip installed in every person. Personal improvement is the new bitcoin mining, called fitcoin mining. The more you work out and study, the more fitcoins you have.
Thus, when Picard says people work to better themselves, they are just mining more fitcoins. All other currency (dollars, renminbi, euros, even bitcoins) are eliminated in favor of fitcoins. Thus you reconcile statements saying there is no money since the late 22nd century with the obvious references to money in TOS etc.
Those money references are references to fitcoins, which people earn through bettering themselves.