Hi Scott,
Thanks for the scene file, it makes it so much easier to tell what is the reason for that.
My first impression is, that it is too dense from the start. You are close to have a model that feels already SubDivided, but it isn’t.
I know that the holes are the starting point for the density of the mesh. There are ways to get the density down, which is work of course but helps to lower the work-load all the way down to the rendering, etc.
The target is always an efficient mesh, not only for render-time, the more points your model “owns”, the harder it is to work on changes. This object has 4608 Polygons, and the Subdivision pushes it on “level 3” to 299.520 polygons. This is certainly huge for a single and simple object like this, which could be seen as an subjective input of mine.
Quads are certainly a good idea, but the ideal is a square, not a super long rectangular. Which could be a “lead” to the problem, but I guess there is something more prominent in the object: Check the image that I have attached. Each second edge along the Z axis follows NOT the circle that it certainly is supposed to follow. If that is only every second edge than the Phong Tag might blur that away (hence my first check on these). With the SubDivision, this change of angles each second time will be amplified, will say the flat part is more pronounced. The angle between two neighbors is calculated along the Phong settings to round the surface—from one normal pin to the next, as an progressive interpolation, soft and smooth. If the SubDivision introduces more edges, these edges will have their own “Phong/Normal” calculations. In other words, the area that was smooth before has now shrunk to an 1/8th of its former space (3to the power of 2 equals 8—the result of Subdivision of 3). In this way the “faceting” of the original surface will be stronger visible. The problem you ask to solve.
In this Multi-gon shape instead of a circular shape I see the main problem. Secondary would be the continuation of the high density of the mesh and the long rectangular polygons. Any detail might work with less polygons.
I placed the object to a XYZ 000 position, and tried to rotate the object, so I would have it symmetrically. As we have twelve holes the angle between two holes is 30º, so I set up the R.B to 15º, but it didn’t aligned, I had to go to 11.3º, certainly not a “biggy”, but a sign that something is off. If you go to the holes and switch to the Line/Wire view you might see even more. Checking the mesh, resulted in 96 non-planar polygons (Edit>Project Info), which is not a problem normally, but I do not see a reason for that, they could be all planar in this model. They are all inside of the hole, so not a concern at all, but an indicator to me, as you asked to inspect this object to find answers.
I do not critique nor criticize here your model or anything thereof, I share my findings. All in all, your ideas how things have to be is not far from being successful, just a tiny little bit more care perhaps to make your work great. You are very close, go further!
As a tip, I show you a little procedural—how to reduce the density while you move on.
All the best
Sassi
P.S.: I used a reference Circle Spline which I had set up to 1º for the next intermediate point, to get the best “reference”.