Would personal servers (with relays running on them) solve the "less syncable" issue? That is, rather than the edge device / client having the chat transcript, an always-on personal server would have this and mediate your chats for you, hydrating your various edge devices with a single chat state

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Perhaps optionally. Like modes for "just keep it on this device" and "keep it on this device but I'm a server and will provide it to other devices when asked".

Assume a secure connection between the personal server and all one's edge devices such that clients act more like remote controls for the "main agent" on the server.

Yup. This is definitely an option for syncing chat transcripts. And it would be implemented at the client/application layer.