I think understand what you’re after but post a model or other reference images if not. It sounds like you want to make a physical emboss of the logo on the flat top of a capped cylinder. Maybe this is for a 3D print. If that’s right, I’d suggest using the PictureFrame command to import the reference image you have now and scale it using the Scale command or the Gumball until it’s the right size. Then lock it and use the Curve command to trace the lines of the letters making sure to use end Osnaps as you go so that you end up with closed curves for each shape or two closed curves as in the case of the D etc. Then use the command ExtrudeCrv with the Solid option enabled to make the physical letters at the height desired. Move them into position and BooleanUnion with the cylinder after that.
You could also try to use the ai file but my experience has been they tend to be open paths not suited to making closed 3D models. You can clean up the curves or redraw them as I’ve suggested… I think the latter is usually faster.
I did this in Illustrator. It is “just” filtering out the outlines. The problem with shaded and shadowed vector art is, you need a lot of objects and layers to get the right appearance.
If the logo were to be a fancy 3D badge with say, a curved front and beveled sides that you want to attach to the cylinder, you could:
Make a copy of the cylinder.
Explode a copy of the cylinder,
Unroll the cylinder developable surface,
Create a flat surface to use as a source.
Place a copy of your logo badge on the source surface.
Flow along surface using the flat source and the cylindrical target.
(You would not want the option to keep the object ridged checked.)
Rather than messing with Pi and math, you could:
Slice the cylinder copy to how big you want the logo, in both directions.
Unroll it,
Resize (a copy) of your logo to the unrolled size
Flow it, as in above.
I’d use the Curve command mostly and make two control points only for the straight lines. Make new curves snapping to the previous curves end point using Osnaps. Any spot with a corner marks a new curve location. Use the Ellipse command with the Deformable option for the ellipses. Don’t forget you can turn on control points and edit the curves too. Join curves together into polycurves too to keep them together while editing.