Besides background syncing of the entire vault, my understanding is that Bitwarden does use a push mechanism (WebSockets) to initiate an immediate sync from the device in which a change was saved:
Please note that when “Live Sync” of individually updated items occurs, the last sync timestamp in Settings > Vault is not updated, so in part, you may have been mislead by this.
I don’t know whether the “issue” referenced by @dwbit would have impacted the “Live Sync” functionality, as well. FWIW, it seems to be working fine on my end (in Chrome, extension version 2025.4.0).