The Stirling engine that runs on wet paper as a cold reservoir is obeying the Second Law, which does apply to it. It's just more complicated, because in addition to air temperature, you have humidity as a factor in the system. Entropy also depends on humidity, and not only on temperature — and in fact you can determine how the entropy depends on humidity, by producing the under-saturated warm air by heating saturated cool air, and keeping the running total of heat input over temperature as the temperature rises. Anyway, evaporating your liquid water is increasing entropy, and that's why this engine works — not because it gets to disobey the Second Law by virtue of being an open system.
It's the 'few drops' of liquid water that are the bottleneck in this system. In this dry air, liquid water is not naturally occurring. Puddles will evaporate away; if not, then the Stirling engine won't work, because there won't be any evaporative cooling. So to make the engine keep on running indefinitely, you need to keep on bringing in liquid water, somehow — for example, from a cooler place, where the humidity of the air around your engine is saturation humidity. That would be your cold reservoir.
Or alternatively you could bring in your water from a place where the air temperature was the same as it is around your engine, but where the humidity was a lot higher. Then it would be true that you had an engine that didn't use a temperature gradient. It would be using a humidity gradient instead. From an entropy point of view, this is just the same. It would work, but it wouldn't be novel, or very efficient. Its power output would not be determined by the large amount of ambient heat in the air, but by the humidity gradient.
The great point of heat engines is that we can create really huge temperature gradients, by heating things up by burning fuel. So we can get huge power out of temperature gradients, because we can make steep temperature gradients. It's much harder to make a really step humidity gradient, so there's just not enough power in pure humidity gradients to be worth bothering with. If you look at all the heat in the air, you think you're tapping this huge energy source, but if you understand the Second Law, you realize that your very limited humidity gradient is only going to let you extract a tiny scrap of that thermal energy. Cancel the IPO. Bummer. That's why you need to consider the Second Law, and not just the first.
It's easy to overlook this seemingly tiny issue of supplying a bit of liquid water, because of course it's easy to turn on a tap and get a few drops of water, in a world powered by city-scale generators. But I have a Stirling engine just like that guy's from the video. It's an unbelievably wimpy engine. It can barely turn its little cardboard wheel. Its power output is extremely tiny. To get any actually useful amount of power out of a thing like that, I'd have to scale it up enormously. At that point I'd notice that I was needing to pipe in water from a cold and/or wet place — or refrigerate the water, or pressurize the air, or something.
Invoking 'open system' does not evade the Second Law. It just makes it easier to kid yourself, by making it a bit harder to see where all the entropy is.
