Why is dimensionality reduction used if it almost always reduces the explained variation? Let's say I have $N$ covariates in my regression model, and they explain 95% of the variation of the target set, i.e. $r^2=0.95$. If there are multicollinearity between these covariates and PCA is performed to reduce the dimensionality. If the principal components explains, say 80% of the variation (as opposed to 95%), then I have incurred some loss in the accuracy of my model.
Effectively, if PCA solves the issue of multicollinearity at the cost of accuracy, is there any benefit to it, other than the fact it can speed up model training and can reduce collinear covariates into statistically independent and robust variables?

In your question there is an implicit assumption about the regressor being linear.
In case it is linear your assertion is correct.
But for the case of non linear regressor you may think on the dimensionality reduction step as a feature extraction.
In that case it has a very important role in order to get good results.
It might reduce the noise, it might assist with the learning, etc...
