Potable water from rainwater - questions, common issues, & fixes?

When I first learned about rainwater catchment systems (metal roof --> big gutter --> big holding tank --> some kind of filtration --> pressure pump --> maybe more filtration --> faucets, showers, toilet, laundry machine, etc.) I figured "cool, those are the parts, put them together correctly and they just work."

Someone on here (John C. Daley, I think?) has said you don't even need filtration. I accept that this person has not needed it, but I wonder if it depends on what's in your local rainwater, also perhaps on the design of certain parts of the system. (How localized are quality differences in rainwater anyway?)

Last night I watched a video by a guy who built an off-grid tinyhouse in Bulgaria or Belarus (some eastern European country whose name started with B, sorry I forget), and he said he used rainwater catchment exclusively for the first few years but got sick of algae taste and smell in his water, and had noticed the walls of his bathroom even started to turn green, so he switched to a well.

How common is the issue he described? What causes it? Did he simply do something wrong in his installation?

I definitely get a lot of algae in my rain barrels, but they are outside and not UV protected so I kinda expected that.

What else is there to say about purity/potability from rainwater catchment?

(Sorry if I'm repeating previous thread/s)