That's probably an overstatement. Of course we could patrol and investigate, and supervise and manage without trainers. But could we do it well, and properly? In many cases, no, we could not.
The trainer takes information from management and puts it into a context that officers will be able to use successfully. The trainer works with officers, absorbing their input and observing their work, and creating a means by which existing protocols can be adjusted so that everything works more smoothly, and officer safety is enhanced.
The trainer is also the "idea person" in the department, continually researching and learning, seeking out new information that officers need, and figuring out how to get it to them in a timely and cost efficient manner. In many respects, most members of the department have a little bit of "trainer" in them. They all share information with each other, and learn from each other.
But the officers who are designated as trainers—be they firearms instructors, driving instructors, FTOs, or any one of a dozen other disciplines—are the ones who have a specific responsibility to garner and convey information effectively. That is a huge responsibility, and not just anyone can do it.
Trainers need to be sharp, eloquent, thoughtful, curious, firm, fair, interesting, honest, and thorough. They need to always be curious about the nature of the profession. They should never feel satisfied that they know all they need to know. If a trainer develops a mindset that he has "completed" his training, he has reached a point where he is no longer providing maximum benefit to fellow officers.