Great work investigating the saponification process, Helena. You are right that there are many simple ways of producing the lye and then making soap by mixing this with a source of oil or fat. I have done this myself and can confirm that it works. There are a few practical things to consider:

1. The strength of the lye. If not concentrated enough the lye will only partially react with the fat and it will end up as a mess and not soap. It would be good to somehow standardize a measure for lye concentration. It is of course easy to concentrate the lye by just boiling off some of the water. By the way, I have also used rainwater and never tap water.
2. Time for the saponification reaction and soap to set. When I tried this process, I just heated up the lye and fat to the required temp, mixed them and then let them sit in my cellar for months as I was not in a hurry. But we will not have months. We should come up with a plan to accelerate the process.

I attached the protocol that I used to make lye.