This could work, but sounds like a lot of work. I think there are more efficenct ways to use rain water. My grand dad is collecting the rain water for years for his garden. He doesn't really filter it and only uses it for his plants, washing and showering and sometimes for cooking. Once you boil it you kill all the bad stuff anyway. It is quite important to have a large area to collect water from. Also there are already filters you could use directly for the water.
For showering, washing the clothes and gardening ok.
But now rain water has toxins from pollution.
Your grandad had not this problem decades ago.
And the filters you mentioned probably are not natural/safe and i prefer to stick with nature possibly.
But we may discuss. Post an example filter please.
Boiling the water I'm reluctant. I've read it screws the water up, due to change in its ions or something like that.
I have still to get info on this, but I'm sure you don't get the most out of the water in that way.