Greywater toilet flushing system.

I'm noodling about a small greywater system.
I have an old but incredibly reliable washing machine that uses a lot of water.
I would like to set it up in the basement and capture the greywater for flushing the basement toilet.
To avoid the problem of greywater turning black I would use a sand filter and a recirculating pump.
The sand filter would take up most of an old water heater tank and it would feed a secondary tank, that in turn feeds the toilet via the usual float va0mlve.

A low flow pump continously would take water from the sump tank and spray it over the top of the sand filter to aerate the water, thus keeping it from turning into black water.

Tap water would enter the system if needed  via a dedicated float valve in the secondary tank.

I am concerned that the clorine in the mains water will impead the biological function of the sand filter, but I think a back up for the grey water is needed.

It could be introduced into the system via yet another tank, and isolated via a one way valve, but I'm hopping to avoid that.

If there is chlorine in the water you can test for it.
Instead of a sand filter an open aerobic system with rocks, oyster shells etc may be better as the medium.

Washing machines use a ton of water. Toilets don't. Why not just use a Japanese-style sink tank for the toilet? After using the toilet, when you flush the water comes through a faucet at the top of the toilet tank, where you wash your hands while the tank is being filled. It's a little more water than you need for washing your hands, but it's a very simple greywater system.

Washing machines also have a pump that pumps out the water. That makes them good for pumping water out to the garden, which might be a better re-use? It would be much simpler than having to clean the water for toilet use.

If I had an old water hog washer, I think I would put my energy into trying to capture the rinse water to use for the next load's wash (but I would not save it for more than a couple hours).  Actually if I had an old water hog washer I would get rid of it and try to find something more efficient... but I live in the drylands, where we can't afford to waste a drop.  Local conditions will impact your design.

But keep it simple... in my mind, purifying grey water (at least without using a planting bed to do so) is not simple...

Good luck!