At first glance, no, this doesn't make sense. You cannot be cheaper than your cost of doing business, otherwise you go bankrupt, non-profit or not. So if you are built on AWS, you cannot be cheaper than AWS. If you use their cheapest tier, you put the same limitations on your customers as that cheapest tier.

The way non-profits offer free or cheap services is not by having a cheaper business model, it is by getting their money from donations. But that means you need to be cultivating donors, always. Instead of having a few major funding rounds with a few investors, you dedicate what will feel like your entire life to finding people to donate small amounts to keep you running. Get enough donors and you can offer cheaper services.

At therein lies the catch - people with money are bombarded with people asking for money all the time. Many of them do give donations, but you are competing with charities who bring good deeds to the world, research cures for pesky diseases and whatnot. Your pitch to them is going to be what? "Instead of curing cancer, give me your money and I'll help people store their media"

In all honesty, the first step is to go dig deep into how non-profits actually work. Because you don't seem to be really driving to run a non-profit, you seem to be looking for a way to sell a wrapper app around S3's glacier tier, and that is absolutely the kind of thing you should just keep doing as an indy dev, not wrap all the bureaucracy and donation development of a non-profit into it.

This is an amazing idea, with many potential use cases, but I feel like, for the layman, a major use case of cloud storage systems like Google Drive and iCloud is the ability to access the same files from multiple devices. This, while being a pretty cool system for backup and restore (and very economical!), is not really as much of a concern for the average user of Google Drive and iCloud, because based on what I've understood, it doesn't allow other devices to read data from the cloud for local viewing. Then again, I might be wrong, and it might be pretty useful for the average user.

You could form a cooperative that periodically negotiates with providers a cheaper bulk price. If the co-op grows into millions of users they could start having their own data centers.

Each person should still have to pay for their usage, but the price could be cheaper.