How does one flush "carefully/slowly"? The speed you press the handle doesn't change the pressure of the flush - at most it might extend it and waste water, by keeping the flapper open longer than necessary.
Instead press the handle rapidly and briefly to save water and still have a proper flush.
If you have C. diff. and don't want to spread it, it's probably worth it to spend (not waste, spend) the extra water for a slower flush if your toilet will work that way.
Instead you'll get the exact same regular violent flush, and then water will pour into the tank extending the time that water can aerosolsize, and worse, pushing any existing aerosols farther out into the room.
So basically that's the opposite of what you should do, if that's your goal.
Instead, reduce the flush time.
Try it: hold down the flush lever and time how long it takes before you can smell the water.
That's not what's being suggested. Slowly open the valve by slowly pressing the lever to let some water in before the valve pop all of the way open.
Plus toilets are designed to prevent you from doing that, they try very hard to be either fully closed, or fully open.
And, even if you succeeded, by releasing some water in advance, you reduce the pressure of the flush, which would elongate it, instead of rapidly removing the material, and time with water flowing = amount of aerosol.
This plan seems counterproductive.