Wouldn't there be anything in Flickr's ToS that would prevent third party developers from hosting their app's photos there? I like the suggestion though.
Yes. From Flickr: "The direct link to a photo file is no longer shown on the page. Per the Flickr Community Guidelines "pages on other websites that display content hosted on flickr.com must provide a link from each photo or video back to its page on Flickr." Linking directly to the photo file doesn't do this."
I think a desktop and/or mobile software for P 360 would make more sense than an online solution. It could simply integrate with people's Flickr accounts and get out of the way.