First off:
That's just it, we didn't. By my count there are three (maybe four if you squint) references to "no money," One in a movie (FC), the rest are episodes. Two of the contention are by Picard, with both he sounds like a starry-eyed dreamer (especially in FC), the other one (or two) are by Sisko. Riker once states that Riker doesn't carry money, but not that it simple doesn't exist in any form.
Starry-eyed dreamer? He was concentrating on keeping an eye out for the Borg while leading Lily down the corridor. Telling her about the future was secondary, in terms of where his attention was; he wasn't "starry-eyed", he was distracted.
When did
Ben Sisko mention the no-money thing?
Jake Sisko did, in "In the Cards" (and I hate that line, by the way), but I don't remember any other references to it in DS9. If they were there, you'll have to refresh me...
Finally, you're forgetting the "no money" reference that predates ALL of those: Star Trek IV. So all those referencing TNG as the source of the "commie hippie nonsense" might want to take that into account.
Why do people always link currency with greed in these types of threads? We don't use money because we're greedy, we use it because it's PRACTICAL. You can have greed in a money-less world just as well. Say someone decides to replicate a hundred boats. Then his neighbour decides to replicate two hundred. Then three hundred, etc. With what did they deserve draining so much energy that might better be used somewhere else? Energy may be cheap, but even in Trek it isn't limitless. How do you limit usage?
Besides, the replicator won't get rid of ALL scarcity. Land will still be scarce. Means of transportation as well.
I always took the "we don't use money" as meaning "it exists, but we don't have to use it in our lives". The Federation provides you with a replicator and enough energy/raw-mass to replicate the basic things you need to live - enough food, clothes, furniture etc. It guarantees you a sufficiently large house to live normally. You have free local public transport. And so on. You can live your whole life without needing to use money. However, if you want a bigger house, or a house in a better location, or to replicate something more energy-intensive, or to get something man-made, or to travel to the stars - you have to work and earn credits.
Excellent post. This is very similar to how I view it. I think that
physical currency is something that the Federation has more or less abandoned in favor of electronic credits. And even that credit system is viewed only as a means to an end, a system by which the value of goods and services can be measured. "Being rich" is, in and of itself, not something that anyone really cares about.
Basic needs are provided, so that no one will be forced to live in abject poverty or die of hunger out on the street (and those whose talents lie more in creative arts of some kind don't have to waste time in a menial business job they have no real interest in, just to have money to survive, while being unable to pursue their art, which happens in real life a LOT). But if you want not just a clean, functional living space, but a nice big house; not just nutritious, decently palatable food, but fancy steak dinners; not just clothing to protect you from the weather, but a suit of fine Tholian silk, then you're on your own.
Besides, replicators aren't simply a magic "make stuff forever" button. You need material to be converted and energy to power the thing, and neither of those are simply endless resources. Some form of economy - even if it's ONLY there to provide structure - is necessary, especially when you consider that even if
humans had abandoned any form of currency or economy completely (which I don't believe to be the case), there are over a hundred other UFP members who may not agree. Not to mention the realities of trading with other, non-UFP powers. You can't just barter
everything.
And all of that said, to some degree, we can't know
exactly how the UFP economy works, ever. For one thing, those who made the show don't even know. It's like the warp drive (and hell, the replicator too, for that matter... and a lot of other Trek tech

): in truth, NO ONE knows
exactly how it works. We have some general ideas, some technical information (most of it fictional, some of it contradictory), but the precise mechanics of the process that begins when someone on the bridge presses a button and ends when the ship begins to travel faster than light can never be completely known. All we can do is look at the limited evidence, decide which of the contradictory data points we are going to run with, and speculate on how the economy might function.