Adding only Cyan will add density, but change the colour of the light because the Magenta and Yellow filtration will have their effect diluted by the Cyan, the net effect will be a lowering of contrast. This is not the effect being requested.
In theory, adding cyan filtration does nothing but remove red light from the mix; blue and green light are unaffected. Because most B&W papers are insensitive to red light, the effect should be nothing (or close to it, given that the paper's red-insensitivity is imperfect). That said, I don't know how "tight" most cyan filters are -- if they actually block even a small amount of blue and/or green light, they'll have some effect on contrast. Thus, it could be that in practice what you're saying is true for some or all subtractive enlargers.
On a practical note, but one that probably won't be useful to fidget, I've got a Philips PCS130 with a PCS150 control module/light source. This is an additive color enlarger -- it's got separate red, green, and blue channels rather than cyan, magenta, and yellow filters. I've done some tests with this enlarger, and I found that completely turning off the red light has no effect on my B&W prints. Exposing using the blue and green lights alone produces prints that are indistinguishable from what I get with all three lights. This is theoretically equivalent to dialing in maximum cyan on a subtractive enlarger -- but keep my above caveat in mind. Also, and more to the original point, adjusting the green and blue lights' brightness by equal amounts affects exposure. The lights are calibrated in Kodak color filtration values, and adjusting both the green and the blue by 30 units changes exposure by one stop (doubling or halving exposure time or changing the lens's aperture by one stop). If your enlarger is calibrated in the same way, the same adjustments should work for you, or at least be close.
All of this applies to B&W VC printing. For color printing, an adjustment of all three filter values is necessary to adjust exposure. In theory, increasing the filter values by 30 will effect a 1-stop change in exposure with no change in final print color. In practice, if the filter calibration is off or if the filters "leak" light outside of their claimed color ranges, there may be a slight color shift when you make such an adjustment. With my Philips, I'm able to make such changes with no change in color. (At least, when the changes are small; I generally only tweak it by a 10-30 units to get the exposure time I want.)
