One of the other great myths about record production, beside the three I mentioned earlier, is that you need to be first and foremost a genius with studio equipment.
This is absolutely not true, because the equipment is only a means to an end. Many engineers have just a basic technical knowledge and would privately admit that they are far from knowing every last detail about every piece of equipment in the studio.
One has to remember too that equipment is designed by electronic and software engineers, not music recording engineers, and although most manufacturers do their utmost to ensure that their products are exactly what recording engineers need (even if recording engineers themselves don't always know how they would like technology to progress!), inevitably most modern pieces of equipment offer a range of functions far in excess of real life practicality.
80% of engineering is not technical knowledge, it is knowing when something sounds right. And what's more, knowing what to do to make something that sounds almost right, exactly right in its musical context. So if you know how to route signals around a mixing console, and you know how to operate the basic outboard equipment, then the rest of it is really down to listening.
Your ears will tell you which microphones to use and where to place them, they will also tell you when to use EQ and compression, and what settings to use. And as you develop your experience they will also tell you when a musical idea is working and when it isn't. This last point is where the boundary between engineering and production lies, and I believe that you don't need to be a musician of any kind to develop from an engineer into a producer along this route.
Any engineer will start by learning the basic equipment operation, and how to spot technical faults in a recording like excessive noise, clicks or distortion. But once a reasonable degree of technical mastery is achieved, then I don't think it is unreasonable to say that you have achieved a musical status equivalent to an arranger or musical director.
You may not be able to bash out a tune on an instrument, but an arranger won't be able to say that if a guitar line isn't working in the studio, all it needs is a bit of compression and chorus (for example) and a slightly different playing technique, and then it will work.
The engineer producer who lacks conventional musical skills will probably work with a band that can supply all the necessary musical knowledge, and translate their work from a brilliant stage performance into an equally effective studio recording.