Many folks here say or imply that getting the "correct" exposure is a must if you want to get good images. Many will add that you have to shoot in "manual" to get control of the camera and get that "correct" exposure. I'm wondering what message this sends to newcomers.

Yes, you should try to get the exposure as close to "perfect" in the camera.

Yes, there are difficult lighting situations that can cause the camera's light meter to give you readings that may be wrong for what you want. But, let's face it, changing exposure is just a matter of letting in more or less light and/or changing the ISO. The light meters in modern cameras are pretty darn good. And if the lighting is squirrelly, you can make the proper up or down adjustments using exposure compensation if you're in one of the auto modes. And you have a fair amount of control in post-processing, particularly if you're shooting RAW.

I don't mean to resuscitate the Manual vs. semi-auto modes debate. I'm just wondering if there is too much of a mystique being attached to getting the "proper" exposure. So how important is it to "nail" the exposure settings? Aren't there more important, or at least equally important considerations such as focus, depth of field, etc.?

I try to get the exposure right FOR MY EYES, not the camera's. I commonly use Aperture priority and exposure compensation to get the exposure right IN MY EYES.
For me the correct exposure may be what other's consider overexposed or underexposed. But, IN MY EYES, it is the correct exposure. The following are some examples of what, in my eyes, are correct exposures.
And yes, I can do work in post, put it is usually for other things besides exposure. And I use JEPG. and I find it works really well for me even in post.

There is no such thing as "correct exposure." It's almost always a compromise between having highlight and shadow detail that both look acceptable. editing your images, and shooting in RAW gives a lot more latitude to being a little off on exposure. The latest full frame cameras are incredible in dynamic range so getting close on exposure is usually sufficient. An image that portrays a somber/dark mood my be better a little darker, whereas a lighter image helps evoke a happy and lighter feeling. It's an artistic subjective choice. If you shoot in manual mode you change shutter speed or aperture to change exposure, in automatic modes you use exposure compensation. The effect is the same, exactly the same. There are people who insist on shooting in M mode, but most do it for reasons they probably don't even understand themselves. Night shots are about the only time I ever use M, especially moon images. Light meters work the same in manual or auto modes. You just correct the image slightly differently. To argue the point I arranged some students in different lighting conditions and asked the M mode users to take a shot of each one as quickly as possible, while I did the same in Aperture priority. I shot all six in the time the students were still lining up shutter speed and aperture manually. You don't get different or better exposure, you just get it differently. I shoot most of my images in Aperture Priority. Another time I use manual is at sunset. As you recompose and the meter reads the sun and shadows in different parts of the frame, the exposure reading changes. I get a good exposure first, go to manual and use those settings, and then changes in composition won't affect the image if I don't read the meter and readjust each time. The moon is similar. Get a good exposure and lock it in using M settings. Unless there are clouds the same setting should work on all images. As the moon changes slightly in framing the meter will give different suggestions, ignore it. Most of my students who insist on shooting in manual do so because of the false belief that it's more professional and makes them look like pros. Also, if you're not editing your images you are sorely missing out on the control you could have.