There's now a strong body of evidence and research that effectively refutes this theory. Basically, human multi-tasking is a myth. Dr. Edward Hallowell is one of many researchers that covers this concept well. He wrote that multi-tasking is "mythical activity in which people believe they can perform two or more tasks simultaneously as effectively as one."
Multi-tasking while working, studying for an exam, or performing most tasks just doesn't work out.
Now, apply this to police or tactical work. Civilians who attempt to multi-task can do so in a relatively safe and sterile environment. Mistakes can often be corrected. Subpar work can be improved with a do-over. Cops, unlike the civilians we protect, have to deal with bad guys trying to kill us. All of our focus needs to be on that dimension. At an active-shooter event, the mission upon arrival is first and foremost to find and engage the shooter. Is the shooter down and no longer a threat or is he still a viable danger? Multi-tasking won't help resolve this issue.
In his book "Just 2 Seconds," Gavin de Becker addresses the problem with multi-tasking. He gives the example of a high-profile protective detail. You can't expect your protectors to be 100% effective, if you try to task them with mundane, collateral jobs such as carrying and stowing luggage or fetching coffee. The mission should be the main focus. The mission at the active shooter event is to stop the killing.
An
active-shooter
call usually doesn't go according to protocol. You most likely will not arrive with an optimal number of personnel for a four-officer contact team, let alone a four-officer rescue team. The site will not always be in a building with hallways and classrooms or offices like your training building.