This sounds like a great case for a decentralized web infrastructure like Mastodon uses. Each cat server can be exactly as reliable as the owner feels like making it, and each user can choose a cat server based on their needs, and the whole thing is federated because it's based on open source and open standards.
mastodon also downloads remote media from all known accounts with no real way to manage it, making it a nightmare to administer disk usage for any decent sized instance. they go the route of your instance being responsible for all remote media, not the other way around
Each server could have an address that you type into a browser. We could have a 'domain name server' that keeps a list of which address should match to which server.
Then people could easily navigate to cat pictures.
p2p cats for teh win