First of all, you need to be careful of your pronouns. "They" the inventor of the device is mostly just saying it can be used to replace satellite thrusters, which would be a huge weight saving (no need for maneuvering or stationkeeping fuel). "They" NASA are saying that, *if* it scales up the way their current model says (yes, they have one), then after a lot of refinement and with a nuclear reactor powering it, this thing could produce hundreds of Newtons of thrust at a scale that would be feasible for spacecraft propulsion. Some people have worked out, based on the predicted thrust and the likely mass of such a craft generating that thrust, that it can be used to reach Alpha Centauri in under a century. The first of these is an obvious use case for anything that can generate a tiny thrust for a long time. The second is a straightforward application of the current-best (though probably still in need of major refinement) model of thrust detected to power input, which has been measured. The model is a guess, but the fact that more power = more thrust has been demonstrated; it is, as you say, a concrete fact. The third thing - the starship drive - is simply once again a straightforward application of the results of the model. Nobody is saying that the EmDrive makes starships possible, just that if we *had* a starship, and if it was propelled by an EmDrive that corresponds to the current model of power output, then it could reach Alpha Centauri in N years. That's simple mathematics; you can do the same for chemical rockets or solar sails or any other form of propulsion that works in vacuum.
Second, I don't know why you're calling the idea of scaling up an observed result "delusional". The first Wright Flyer could barely get one person off the ground for a few seconds, flew slower than a horse could run, and was so fragile that it was shattered by a gust of wind after only four flights. But, once you've demonstrated that it *works* - that heavier-than-air flight is actually possible (which should have been obvious to everybody, given that birds and bats and insects exist, but plenty of people thought humans would never achieve it) - then scaling that up to WW2 bombers was pretty straightforward: more-powerful-for-their-weight engines (the Wright brothers had to design their own engine; the existing ones at the time didn't have enough power to weight ratio), stronger-for-their-weight materials, lots of refinements to the design (the Wright brothers pioneered the use of wind tunnels and used them to fix several significant errors in the equations that govern aspects of flight like lift, but their first Flyer was still a very primitive design with many compromises or outright flaws), and other simple, iterative improvements. Breaking the sound barrier was a bigger challenge than getting across the ocean in one flight, in terms of theoretical challenges. All this from a craft that could barely get one guy a few hundred feet down a beach.
If somebody had watched the Wright Flyer and said "one day, people will be able to fly to Europe from the US in less than a day" would you have called them "totally delusional" too?