Does this process ever becomes a second nature?

You've got to be able to instinctively know how to play a position for blitz/bullet chess. You don't have time to ponder over tactics, so you should be able to do those things instinctively(you can check your move with calculations in rapid chess, and if you have time in blitz).

If you are interested in the psychology of learning and improving in chess, I really recommend reading Chess for Zebras from Jonathan Rowson (and/or Move First Think Later from Willy Hendriks, which has a very similar message).

tl;dr: Yes, finding the "best" move continually becomes more and more natural (and thus faster!) while you improve (although even masters never reach the point of perfection).

Rowson explicitly distinguishes between Knowledge (theory learned from books etc.) and Skill (the practiced ability to actually use that knowledge on the board). If you think about learning languages, the former would be knowledge about grammar and vocabulary (you can learn that to a reasonable degree in school), and the latter would be the ability to speak/write fluently (you can only really learn this by constant practice of the language). Someone with only school knowledge will have to consciously think about everything and thus needs more time than a skilled speaker to form sentences - even if he/she knows all the parts!

It works roughly the same way in chess, up to the point that strong masters can often subconsciously "feel" a move within seconds and then "just" have to use their calculation abilities to make sure it actually works.