Fans have spent nearly four decades techno-babbling up Star Wars to fit what we see or are told to get around the as you say "Spaceships go Vroom! Pew! Pew! pew!" so what is to stop Star Trek fan from spending time trying to eco-babble the Earth or Federation economy into something that functions without money by the late 23rd century, and improved to work even better by the 24th century. To get around the "economy goes Vroom! Pew! Pew! Pew! We don't need no steenking moneys!'
Seems only fair.
The problem is the fan attempts to technobabble up economic explanations fall about as flat as the classically awful technobabble of Star Trek medical episodes. For much the same reason. The writers were writing future technobabble about an area that they clearly had no real working knowledge of, beyond what they had seen on another tv drama. That's the strange thing about Trek. They often had science consultants, even occasionally people from NASA to help them flesh out how things would work or need to work with regards to the ships and technology. And it worked perfectly within the narrow confines of their living and working in space viewpoint.
But it often fell apart rapidly when they touched on other areas of life. What the impact of various tech would be? Typically because the writers and creators had limited intimate knowledge of those. Anyone who has any background in actual medicine laughs at Star Treks medical episodes because of how absurd they are. Above and beyond the Deus ex machina technology. They loop around and contradict themselves and actual biology so horribly at times that there is no way to logic up a technobabble explanation that works.
The Star Trek economy is similarly flawed. As others point out they say one thing but actually present another, without the writers or production staff ever realizing it. Gene Rodenberry was particularly horrible for this. It is clear his actual economic education and knowledge was slightly lower than that of a geriatric poodle. (Or slightly above average for Hollywood. Call him a 4.5 on the Kardashian scale.)
See here's where Gene's futuristic "we have no need of money" falls apart. We see property in Star Trek. We see it all over the place. Personal Property. Land ownership. Trade rights. Borders. Mining Rights and materials. We see it everywhere. Heck in early DS9, years before Jake and Nog's classic discussion on money in the Federation (where Ron Moore subversively skewered Gene's moneyless Utopia) we find Jake, the no money Federation communist if you will, teaching Nog to invest in land on Bajor with their surplus trading windfall, as he had been taught by his father and grandfather that land was a good long term investment that holds value. The Piccard family winery was property. Kirk's Iowa farm. Property. The various gifts and antiques and antiquities exchanged on screen. Property.
Property = Money, Money = Property. Currency is simply a common unit for the valuation and exchange of property, goods and services. It doesn't matter if you tie it to a gold or Latinum standard. It does not matter if you call it capitalism, communism, whatever. At some level that exchange must be happening.
Clearly the main takeaway from Star Trek is in the Federation the near infinite energy resources have so lowered the requirements or costs needed to obtain the basics of life. Food, shelter, basic transport and communication, so as to make them nearly invisible to the consumer. So what we consider to be a struggle for existence poverty is pretty much eliminated.(much as it honestly has been in the Western World. We have sadly so redefined poverty to now mean "not as many consumer goods as someone else.")
Here is perhaps the biggest examples of how we know there is in fact a working economy in the Federation, in spite of what Picard says.
1. Hardcore Fenton Mudd - pay attention to Mudd in his various appearances. Criminal though he be. He illustrates more of the realities of Federation life than I think Gene intended. Similarly watch Cyrano Jones and to a far lesser degree Okona.
2. Mining Contracts and Trade - I started poking through the episodes of TOS and TNG looking for examples of economic activity and commerce. I really wasn't expecting what I found. (I didn't bother with DS9 or Voyager for this as DS9 was clearly showing a broad spectrum of such activity from the word go, and Voyager was by definition cut off from their home economy and forced to scavenge and trade.) in actually looking at the full range of episodes I think we may have some flawed misconceptions about what the Enterprises actual role was. Particularly Kirk's mission. We think it was "exploration", but when you look at the shows in a broader way it suddenly jumps out how much of the mission was securing mining rights and trade agreements. Settling trade disputes. Haggling for and acquiring resources for the Federation. I'm not going to lay them all out here. I'm just saying go look for them. Once you do it leaps out at you. It really is all about the Dilithium.
3. The Apple - when looking through episode lists for point 2, I stumbled across this fairly early classic one. I ended up rewatching it with an eye towards this discussion. Kirk actually discovers a society that is essentially the Utopia that Picard later describes the Federation to be. The natives are ruled by Vaal, essentially a giant replicator. They feed energy into the machine. The machine tends to all their needs. The people are clueless happy morons that have lost even the will and knowledge to procreate. Needless to say Kirk destroys their Utopia. It's a rather shocking contrast to some of the later TNG ideas.