Does anything like this exist? A feed of unfiltered, unmoderated content + conversations, fed to a multitude of websites, each with its own moderation filters and content standards. All submissions go to the to the shared feed, but hit moderation and content filters on the website before being displayed. So you can have a reddit-like website with a lot of good content, but if it ever goes south people can quickly flock to a peer.*

Is this how Mastodon and Nostr expect things to work? If so, are there any "winning" websites where most Mastodon and Nostr users congregate?

For most users the back end philosophy is only there to serve the end result. For any given post or comment you need a simple website link you can share with friends; no thinking required.

*Bonus: if people feel like their views are being improperly censored they can, again, jump to a peer with different filtering standards.