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?

Data reduction (unsupervised learning) is not always used because of any hope of wonderful performance, but rather out of necessity.  When one has the "too many variables too few observations" problem, the primarily alternatives are penalized maximum likelihood estimation (ridge regression, lasso, elastic net, etc.) or data reduction.  Data reduction, which as a side benefit deals well with collinearity, can be more interpretable, and works in any predictive context.  Data reduction is IMHO much preferred over variable selection, because in the majority of problems variable selection yields a result that is too random/unstable.  The spirit of data reduction is this: Estimate the model complexity that can be supported by your available sample size.  Reduce the dimensionality (in a way that is completely masked to Y) and fit a single model whose number of parameters (that are estimated against Y) is supported by the effective sample size.
When using variable clustering or sparse principal components, one represents groups of variables with scores.  Sometimes an entire group can be dropped.  This procedure is not distorted by collinearities.
