This case appears to be one where there's no fault of personnel or procedure; just a bad set of circumstances that case together.
Without being able to see the runway environment, there's no way the controller could confirm the call. It could've been from any aircraft within his airspace, even one he wasn't working. Depending on his workload, he still likely had other traffic to work. When 5912 no longer makes contact, and is shown to be stopped on the runway, that will insure that someone knows and from then, a positive call can be made to the crew querying on what the problem is. It worked out for the best, it appears, because of the items in-place to assist in learning that 5912 was indeed the emergency aircraft and the source of the clipped transmission. I pretty certain that, workload permitting, the controller would've queried over the freq; but it sounds like the freq was pretty busy.