But isn't this what we want. I mean it saves us from the trouble of
multicollinearity isn't it.
Yes! and no. Elastic net is a combination of two regularization techniques, the L2 regularization (used in ridge regression) and L1 regularization (used in LASSO).
Lasso produces naturally sparse models, i.e. most of the variable coefficients will be shrinked to 0 and effectively excluded out of the model. So the least significant variables are shrinked away, before shrinking the others, unlike with ridge, where all variables are shrinked, while none of them are really shrinked to 0.
Elastic net uses a linear combination of both these approaches. The specific case mentioned by Hastie when discussing the method was in the case of large p, small n. Which means: high dimensional data with, relatively few observations. In this case LASSO would (reportedly) only ever select at most n variables, while eliminating all the rest, see paper by Hastie.
It will always depend on the actual dataset, but you can well imagine that you don't always want to have the upper limit on the number of variables in your models being equal to, or lower than the number of your observations.