Maybe a combination? Even simple beamforming/stereo would be helpful to help display the speaker's location. For example, the "speaker 1" tag could appear on the left, center, or right of the display to give a spatial clue where they are located.
The thing you could do is train a localizer on the separated audio. Phase is estimated by the source separation process, so you can actually train an ML model provided you have some ground truth (e.g. estimated human locations from camera detections)