With the patch for #16488 (moved) (rebased to the ESR45 codebase) applied, the "Sign in to Sync" text label is correctly hidden but the icon is still visible in the hamburger menu.
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If you pay primary attention to hamburger menu, then don't forget about Synced Tabs that can be added through Customize in it. The rest is in #16778 (moved).
Looks good to me. One nit: could you wrap that long line in torbutton_update_sync_ui()? It would make it a bit easier to read in my terminal.
Maybe also add a try/catch in that function or check that the getElementById() succeeds. That way a failure there due to future Firefox changes won't break other Torbutton features such as the circuit display.