Perhaps you are using another definition for a covariate, but covariates are meant to be measurable, uncontrollable factors in an experiment. Using covariates in an experiment is a way to handle noise that is measurable in the experiment. The covariate is a random variable in an otherwise fixed effects model (hence mixed model). Analysis of the covariate modifies the data to account for the covariate effect before analysis of the fixed effects model is done. Depending on the size of the experiment, there is a limit to the number of covariates you can account for.
Now you can certainly do regression using random variables all you want.
"All models are wrong, some are useful" G.E.P. Box