UPD: for anyone facing the same issue. Most likely PipeWire for non-audio use-cases was in active use at least from Debian 11, or even earlier, and it totally makes sense to leave it installed and enabled in Debian 12 too. So to resolve the problem it was enough to follow these steps:
1. Ensure that pipewire and pipewire-media-session packages (without recommended packages) are installed. 2. Their services are enabled and running. 3. Reboot (optional). The PipeWire error log entries mentioned in the ticket description should no longer be added. Apparently, if the wireplumber session manager is used instead of pipewire-media-session, it'll try to take over the responsibility for audio subsystem as well. And, of course, it's not what we need, since the goal is for PulseAudio to keep fulfilling the respective role. As a side-note, after PipeWire issues, like the ones described in this ticket: https://gitlab.freedesktop.org/pipewire/pipewire/-/issues/3284 , are resolved in new versions of Debian packages, such steps as well as PulseAudio presence will no longer be necessary. Yura
