>Cold Dark Matter, as used in simulations etc, usually has six free parameters.

That is not correct. LCDM as a cosmology is described by 6 parameters mostly for CMB analyses, but this is much more than just CDM. These parameters include the amount of normal matter, the cosmological constant, the amount of dark matter, reionization, the Hubble constant and two parameters which describe the initial fluctuations. CDM only has one parameter in the model, its density. As you can see there aren't many nobs to turn. These parameters are also fixed to observational values. And if you think you can fit all modern cosmological data with a physically-meaningful model and fewer parameters then go ahead.