refactor: move sync logic out of Client #9824

Job	Run time
Protobuf Backward Compatibility	11s
Cargo Nextest (Linux)	3m 16s
Cargo Nextest (Linux Nightly)	2m 36s
Cargo Clippy	2m 44s
Cargo Fmt	22s
Protocol Schema	3m 14s
Backward Compatibility	2m 43s
Database Migration	2m 33s
Genesis Changes	2m 32s
Sanity Checks	2m 34s
Style	34s
Upgradability	2m 32s
Cargo Audit	25s
Cargo Deny	33s
Non-default Configuration Builds	1m 20s
Themis	40s
Unused Dependencies	2m 31s
Rust Doctests	2m 31s
Cargo check publishable packages separately (Linux, linux, ubuntu-latest)	3m 12s
Cargo check publishable packages separately (Windows, win, windows-latest)	1m 53s
spellcheck	28s
Generate Coverage Artifact	1s
Upload Coverage	1s
	39m 26s

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