Here's how it works:
1) When you turn on hot water, the water starts flowing through the heater. The motion causes the flame to turn on, and the water is heated at that point. I'm going to call that the heat point.
2) There is a legal limit to how hot the water may be, so when the water at the heat point exceeds that, the gas is shut off.
3) If you turn the water off briefly, the new flow causes the flame to turn back on. It does *not* turn back on until the hot water has been shut off.
You have to balance these three principles, and they can vary a lot!
If you turn the water down, so that you aren't getting as much water at once, the water sits longer at the heat point, and gets hotter. Then it goes cold. Why would this happen?
Well, if you are using a trickle of hot water, say to rinse your dishes, it will go cold because it has gotten too hot at the heat point and is no longer heating.
You are also using a small volume of hot water when you run a load of clothes on the 'warm' setting. Your washer will reduce the amount of hot water and add in cold. Again, due to the reduced flow, the heat point gets too hot and it shuts off. You end up with only cold water in the wash.
Now suppose that you are washing - your hands or whatever - and it's too hot. So you mix in some cold. Your home has only just so much water pressure. Turning on the cold water reduces the amount of water going to the heater and - guess what? - the reduced flow causes overheating and it shuts off. (But not until it has heated up a lot, so that the water is now *hotter* than before you turned on the cold.)
The only way around this is to use only hot or cold water, never warm.
Well, why not?
In the first place, suppose you set the temperature to be comfortable for a shower. This means that you can never have the water actually *hot*. I don't shower - I use the bathtub. If you put this moderately-hot water in a tub, it will be cold within about 10 minutes. The tub absorbs a lot of the heat, and because of the large surface, a lot is lost into the air.
So this would mean that I never get a hot bath. And it doesn't improve the laundry issue at all. The hot water will still be too hot for permanent press clothes, and the 'warm' setting will still result in cold water.
And I don't know about you, but I want to wash my socks in water hotter than people want to shower in.
On top of all this, suppose you are in the shower, happily sudsing away, and someone flushes the toilet?
Rush of cold water, reducing water flow at the heat point, overheating, shut off of hot water. Person in shower gets sudden cold. Water has to be shut off, then turned on again to restart the heat. Besides that, all the water from the heater to the shower head is now cold and has to be run out before there is hot water again.
All of this might actually work for some people. If you always shower and use a laundromat; you're not interested in steaming your face at night, or having very hot water to rinse your dishes; you have only one bathroom so that no one is flushing the toilet, and your heater is close to the bathroom so that getting fresh hot water to the shower head is just a couple feet, then you can probably get away with this system.