Does notation ever become "easier"?

As others have pointed out, it gets much better if that's your first semester. But in my experience, there is not much relief between, say, years 2 and 4 of your studies. Sure, you get more mature, but the material gets more difficult too.

To address your question whether "more advanced mathematicians also struggle to extract the ideas from the notation", I'd like to quote V.I. Arnold, since I think it's exactly in the spirit of your frustration.

It is almost impossible for me to read contemporary mathematicians who, instead of saying "Petya washed his hands," write simply: "There is a $t_1<0$ such that the image of $t_1$ under the natural mapping $t_1 \mapsto {\rm Petya}(t_1)$ belongs to the set of dirty hands, and a $t_2$, $t_1<t_2 \leq 0$, such that the image of $t_2$ under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence.''

The trade-off is clear: without rigor math would've been quite a mess. But if rigor is the only way math gets communicated to someone, this person simply won't have time to get far in math.

The question you need to ask yourself is: "If a lecture contained only ideas and intuitions without rigorous formalities, would I be able to write formal proofs on my own?"

If the answer is "yes", then congratulations, you are very promising young mathematician. If it is not the case, you are in the same boat as most of us were when we began studying mathematics (you might still be promising young mathematician, though).

The thing is, formal language is a necessity when we want to clearly state our ideas, check if they are correct and share them with others. Sure, two experts in a field might not need to be very formal when communicating with each other and still understand perfectly what they mean and you listening on the conversation might not understand a single thing they said. If you were interested, you'd want them to tell you the appropriate formal definitions and theorems involved. You might want to know about particular details in some proof and how does one reach desired conclusions. This would not be possible without formal context. Then again, even two experts might find themselves in disagreement in which case they would go back to being formal to clarify things to their satisfaction. Lack of rigor can obscure subtle errors and we've all fallen prey to it at some point. Hence, it is essential for any mathematician to be able to read formal proofs and to write their own.

The beginning courses in mathematics ought to teach you rigor and formal language in order to prepare you for understanding advanced topics. You might struggle now, but when you grasp it, you will be very thankful for it. The goal of learning proofs will shift more to understanding ideas involved and not the formal language, but at that point you should be able to write down ideas formally and judge their correctness by yourself.

To be fair, I agree that some authors are overzealous with formal language while disregarding intuition and ideas. But, this is what the faculty is for, professors and teaching assistants are there to explain and clarify. When in doubt, you should ask for their help.

If its your first semester taking a proof based class, then its very typical that it takes a long time to absorb proofs. The more math classes you take, the more fluent you'll become in reading and understanding proof. It gets better!