You may see there's something like dot rings formed as interference results. CMYK printing tries to avoid them but here they are the wanted result.
The interference happens between equally spaced color dot lines which are tilted to different angles.Every color has its own parallel dot lines but every color has its own line orientation angle. Here the angles are selected so that easy to see repeating patterns are formed. A part of the colors have got blending mode multiply to get also mixed colors where the dots overlap.
The dot sizes in your example obey what's needed to fill a circle with radial gradient. If we forget for a moment the circle we can see those interference patterns in my simpler example:
I have 4 groups of lines of dots which are made with blending.
NOTE: The different sizes of the dots do not make the interference patterns, they are created only by the equal spacings of the dotted lines and the equal spacings of the dots in the lines when there's non-parallel groups of lines. We call it "Moire effect".
As said, in CMYK printing the angles of the lines are specially selected to avoid too offensive Moire patterns. In CMYK printing another as important criteria for the line orientations is good ink distribution on the paper.
There's no automatic way to produce the wanted half-toned radial gradient as vector in Illustrator if the dots must be regular circles and no extra software is available.
I could make a grayscale radial gradient say from 40%k to 0%k and apply to it the halftoning effect that you already have tried. Expanding and tracing it (ignoring white) would give to me a vector pattern that I could copy and make differently colored copies. Giving to one or more versions blending mode multiply I would get also mixed colors.
The on-tracing based vector half-toning is shown in numerous Youtube videos like this https://www.youtube.com/watch?v=feyMOhp5qtE
Unfortunately tracing those quite small rasterized circles generates ugly irregular shapes, not clean vector circles. To get nice looking circular dots the half-toning of the radial gradient must be done as vector. There's commercial plugin Astute Graphics Phantasm which creates vector half-tonings. I do not have one, so I cannot show it.
Not asked: Inkscape has perfectly well working vector haftoning. It's a quite well hidden auxiliary opition in tiling a dot pattern. It's tab "Trace" in the Edit > Clone > Create Tiled Clones-dialog.
It's used in the next image to create a half-toned version of the underlying radial gradient circle (the tiled pattern was a square, but the excessive circles are deleted manually):
The clones are unlinked and grouped. Cyan, yellow and magenta copies are made of the group. Every group got different rotation angle and layer blending mode - no selection rule was used, different rotations and blending modes were tried until the result looked colorful enough: