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.
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.
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.
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.
The chlorination process is simple: just pass some current through the salt and voila, chlorine.
Admittedly, there is a bit of a psychological hurdle to washing your hands in the toilet, even though the water is isolated and perfectly clean. (This could be solved through clever plumbing, to separate the two fixtures.)
Example: https://www.reddit.com/r/mildlyinteresting/comments/9rabuk/t...
The reason so many 'amazing' idea never take off in houses and why cookie cutter housing rules.
There are big regulations, if you cross the piping you could theoretically contaminate the streets or larger water supply.
You now have two water systems in a house. Plumbing is not cheap, having two is more $.
You'll need a pump and holding tank, a place to run the pump and holding tank. It'll might me loud and can break down, will fail in blackouts. Can the tank get legionnaires disease?