Make FallbackManager internal to ReconnectProtocol and communicate with it over the wiring

Currently FallbackManager is a shared part between sync and reconnect logic. To achieve better separation of those, it should be owned by one of these parties (Reconnect is suggested) and other side should communicate with it over the wiring.

Start by locating FallbackManager, ReconnectProtocol, and the sync and reconnect wiring. Trace how both sides currently share FallbackManager, then confirm that ReconnectProtocol owns it and the other side communicates through the wiring; the separation should be covered by the relevant existing tests.

Written by the indexing model from the issue text.