Honestly, I do. I've placed a bucket in the sink to capture waste water from washing hands, which is then reused for flushing the toilet after depositing... liquid contents in it. It saves a bit off the water bill and also made me realize just how much water I use up when washing hands...
Of course I hide the bucket when guests come, it's too weird a thing - but in all honesty, I've wished many times for a system for capturing sink & shower water in a tank, for use in toilet. The standard practice is inefficient.
EDIT: I didn't know the term "greywater" before, and after reading https://en.wikipedia.org/wiki/Greywater I'm having second thoughts about my scheme. If the city is actually treating greywater and water from toilets separately, then I guess it makes sense to let the sink water drain. But then again, that's a huge if. I'll need to take a closer look at my indoor plumbing and read up on the water treatment scheme in my area.
As of now, very few buildings have separate grey and black water plumbing. In the event that you did have separate plumbing in the building, you would need separate plumbing in the street as well.
Unless you live in a very water constrained area, it's pretty unlikely. In the US, plumbing for reclaimed water is available in some areas (purple pipes), and seems like an easier intervention, it's often used for landscape irrigation, which has easily accessible plumbing to retrofit. It can be used for toilet flushing as well, if that is plumbed separately from other fixtures in the building.
Yes, depending on the detergent content. Reclaimed water is usually water that's been treated for pathogens at the treatment plant, but hasn't had been through all of the post-treatment cleansing required before discharging into waterways.
It's exceedingly unlikely your American home is keeping graywater separate from blackwater. If it were, you would know, it would have been mentioned as it's quite exceptional.
California not long ago changed its laws regarding graywater handling in response to the sustained droughts. Now residents are permitted to divert graywater themselves for other purposes like irrigation and gardening. So it's become a little more common to find diverted graywater systems in response to this change, but they're often ad-hoc diy setups in drought-stricken regions.
You're right though, showers and laundry are huge wastes of water, at least relative to residential water use. Unlike toilets, they're also much harder/impossible problems to solve without the voluminous use of clean water. With toilets there are options like composting and incinerating, and in places with ocean access seawater can be used for flushing. Bathing and laundry don't really have alternatives.
Here in Belgium it is mandatory (for quite some years) with every new building development to install a tank capturing rain water from the roof. This water must be used for flushing toilets and you typically also use it to water the garden, wash your car, etc. With some extra (carbon) filters people also use it for their washing machine.
I have a 10000L tank which is more than sufficient for a household of 4 and some garden. The minimum size is calculated based on your roof surface but normally people install a bigger tank. The extra cost for a few 1000L more really isn't that much and it can save you the cost of a fallback system in case the tank runs dry.
The overflow of the tank goes to a separate sewage system that usually ends in a local stream. This way it doesn't dilute the actual grey/black water which, incidentally, makes a water treatment plant more effective (or so they say). It also allows the water to infiltrate into the ground and there's less risk of flooding the streets during a downpour. At least, this is the theory. Work is still very much in progress to split the sewage systems but we're getting there.
You can put a small sump pump with a pressure switch in the bucket, and an overflow setup back into the normal drain. This way when the toilet flushes, the sump will kick on and fill the toilet and keep pressure on the water line feeding it. Any excess sink water will overflow out the upper hose on the bucket and go into the normal drain.
Correct me if I'm wrong, but couldn't you just have the one output from the pump going to the toilet tank? Then if the overflow switch triggers the sump pump, it'll overfill the toilet tank, which will drain into the toilet bowl, and down the drain... right?
Of course I hide the bucket when guests come, it's too weird a thing - but in all honesty, I've wished many times for a system for capturing sink & shower water in a tank, for use in toilet. The standard practice is inefficient.
EDIT: I didn't know the term "greywater" before, and after reading https://en.wikipedia.org/wiki/Greywater I'm having second thoughts about my scheme. If the city is actually treating greywater and water from toilets separately, then I guess it makes sense to let the sink water drain. But then again, that's a huge if. I'll need to take a closer look at my indoor plumbing and read up on the water treatment scheme in my area.