Can we have a federated service that's resilient against outages such as this, where Element would automatically use another server with my same account?

Perhaps the network of servers could even have some redundancy for the last two weeks history of chat messages, minus images.

Matrix's federation already does result in redundancy of the entire chat history; users not on @matrix.org can currently communicate between each other just fine even on :matrix.org rooms despite matrix.org being down.

It's entirely just user accounts that are tied to a homeserver currently. There's a proposal to make it possible for clients to fully manage their account identity (https://github.com/matrix-org/matrix-spec-proposals/pull/408...) but that doesn't look particularly active.