The statement seems right to me.
Using sites as a random effect in a mixed model does not consume any degrees of freedom. In contrast, using sites as a dummy variable in the fixed part of the model, would consume a lot of DF (21-1).
However, you may want to check if sites have any effect at all. If there is no inner site correlation, you may assume that the observations are independent. In this case you could use all observations within the model without a random effect.
Graficcaly, you could check this with a boxplot of the residuals of the model for each site and compare the medians. Or you can compare the AIC of the model with and without the random effect.
