You are probably going to think this is just too gross. I agree that it can be gross and it's texture and odor are unpleasant. We believe that because of the way we've been taught. For a great many people who live in poverty poop is just a part of life and "no big deal." As a child neglect investigator I've been in homes where you have to watch where you step because of the animal and baby poop on the floor. I could smell it but the occupants couldn't. And they seemed to be no more sick than the rest of us who don't allow poop on the floors. That does not mean that I approve of poop out of the toilet, just that having one's baby poop in the bath water is not unhealthy.
When my grandkids did that and the poop was solid I immediately scooped it out and went on with the bath. No need to change water, sterilize the tub, etc. My grandkids did it perhaps 3 times each.
What makes poop unsanitary is for the baby to have an intestinal illness. The poop then has bacteria or virus in it. These germs are harmless until they have sat for awhile giving them time to grow. However, everyone in your household has already been exposed to your familiy's set of germs. The germs in the poop are the same ones to which you've already been exposed.
Besides the poop won't be in the water long enough to dissolve. I suspect that if you tested the poop immediately after it's been passed you'd find less germs in it than on any other place in your bathroom.
The big push to washing hands after using the toilet is related to major illnesses such as hepatitis, worms etc. which your family doesn't have. Yes, it's good to wash your hands after using the toilet but it really is no big deal unless someone is sick and you do not want to pass germs along.
We wash our hands before meals because our hands pick up bacteria from the things that we touch, that others who are ill, have touched. Touching the ground is a way to pick up germs and possibly worms. That is when we wash our hands before using the toilet.
If we don't make washing hands a habit we are likely, especially for children, to not remember to wash them when we need to do so.
As a society we have placed so much concern about staying away from germs that we are unable to build up a resistence to the ones we inevitably come in contact with. Just as we now have ineffective anti-biotics because the bacteria has become resistent to them we are now having more sick kids because everything in their home is so sterilized that their bodies have not built up a resistance to the common germs that exist everywhere.
I'm not saying we should not keep things clean just that a bit of poop quickly removed from the bath water does not mean that we have to empty the water and sterilize the tub.
My grandkids only pooped in the tub 2-3 times each. Perhaps keeping the situation calm and normal helped them not to do it for the excitement. I did calmly tell them poop goes in the toilet as I put it in the toilet but I didn't scold and I didn't expect them to be doing it again. The situation is one of those it's "no big deal" issues.