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?

Take a simple example of computing seasonal adjustment factor for months across a set of years for a company's sales. Assume there is no linear trend except if years are associated with an inflationary period. Note: In reality, one would work a log transform of the data which assumes a constant percent change relationship across time.
Collapsing the month data across years produces good results by month if inflationary periods are rare. If you happen to guess that the year seasonality is in a non-inflationary period, you have the best estimates with the best error estimates. So, the dimensionality reduction (ignoring years) is clearly best.
However, if it turns out that you are in an inflationary periods, not so good monthly seasonal adjustment. However, a year model may capture the inflation trend and produce better results.
So which model to use, collapsed or full?
One approach is to estimate the probability that you could be an inflationary period based on history,
Next, what is the operational cost associated with having an average error of X in a months' seasonality.
Knowing the difference in cost by month due to modeling error for collapsed vs full for inflationary versus non-inflationary, and the associated probability of each case, one can make a decision that produces the lowest expected cost.
This assumes that this exercise repeats over time and starting parameters estimate are good estimates.
So, the specific answer relates to the nature of the data, model specification/estimation precision and associated knowledge relating to error cost estimates.
