Let's say an average American household spends 10716 kWh annually, 893 kWh per month, 30 kWh per day, 1250 W/hour . If we would like to generate that electricity via solar energy, we would need 25 conventional 250W solar panels working approx. 5 hours a day (see calculations here). It's not a cheap game and we definitely need help and here is where your idea could turn out to be very smart!
Let's put down some numbers. In my city, an average of 917 mm of rain falls every year . If 10 mm means we get 10 liters per every square meter, an average of 917 liters of rain drops on every square meter of my city. For the sake of the calculation, let's say an average 200 m2 house has approx. a 140 m2 of roof . That means the roof can collect 140 m2 x 917 liters = 128 380 liters of rain annually!
To generate power of 10716 kWh/year, which an average household spends, we should build a system (in my city!) capable of generating 0.0835 kWh (83.5 Wh) per liter of the rain. That's the amount needed to fully power the household, which is not our primary goal. Let's first see how much can we generate with an existing solutions.
How much can we generate?
I'll use the example of the cool youtube guy you linked in the session description. He dived really deep into solving the issue of rain gutter energy generation.
In the first video, he calculated that he has a flow of 1.87 gallons/min, or around 7 l/min through a single gutter. He calculated it would result in 2 watts of power, if everything is turned to electricity. With power lost along the way and some adjustment to the system (different turbines, siphons and flow controllers), he came from 0.191, up to 0.612 W and ending in 0.819 watts of power in a second. That means that we are generating approximately 50 W/min, or 7 W/liter, right? (correct me if I'm wrong). We are still generating 12 times less power than we need.
How could we enhance the energy generation?
Multiple turbines on different hights. This would probably not work since the highest turbine would generate least energy because of small height difference compared to the roof. The gains would be minimal, especially when we count in the losses due to the water outflow through the prior "leak holes".
Multiple turbines on multiple gutters. Since we calculated the amount of energy per liter, we would not get any extra energy if we installed more gutters with turbines. We would just get it faster. It could be used to instntly charge stronger devices maybe. Instead, with the adjustment of a youtube guy to control the flow of the rain via syphon, we could do a roof redesign and use it as a rain accumulation pond and deliver stable energy even through times with no rain.
Enhance the turbines. I don't have any bold ideas, since the guy did an amazing job in trying to enhance the energy generation from the rain by changing the turbine system, the propellers, etc.