It would seem that the use of Cyan in the enlarger head is there for density use to control the time of exposure, this is the only reason I can think of it being used.
You are correct with all things, except I am not so sure of how the analyser determines the correct exposure.
One thing to understand with colour printing is that there are three colours you are mixing. You obtain two colours by using filters in your enlarger, the third colour, red, is obtained by exposure. Sometimes you will not see the slight red changes, unless your filtration is correct, or nearly correct
If your print is a little too dark it will be slightly redder, if it is a little too light, it will be slightly more Cyan.
Many industrial enlargers and some amateur ones (Meopta for instance) have a neutral density (ND) filter built into the colour head. The idea is to aid in printing by using a neutral grey to allow you to use more or less exposure without changing your exposure time.
My current enlarger does not have a ND filter, so when enlarging colour (some years past now) I would start with all three dials C, M, Y, having equal amounts of filtration, say 30 units each out of 200 possible.
This meant that if the correct filtration should be 70M 35Y then the enlarger head filtration would read, 30C, 100M, 65Y.
If you have made a successful print with this filtration set, then decide to do a doubling of the print size (a bigger enlargement) you have some options. You can make the exposure longer, or open the aperture wider, or reduce ND to make a print with the same density that is correct for you.
Your calculations may tell you that you require 1 stop more light to keep your correct density, if so, these would be the likely ways of obtaining that extra stop of light. Add one stop of exposure, or open the aperture by one stop, or reduce the ND in the colour head by one stop.
To reduce the colour head filtration you would go from 30C, 100M, 65Y to a setting of 0C, 70M, 35Y. Then after you have focused, you use the same time and aperture setting as used with the smaller print and you should end up with a near identical print colour wise.
I believe this may be the concept behind the use of Cyan in the colour head with regard to density, or time of exposure calculations by your colour analyser.
Mick.