From my limited experience, many players and DMs seem to get things backwards in exactly the way you're describing. They take the rulebook as the starting point or the "controls" for the game and since combat is the most detailed they tend to focus on that to the exclusion of other parts of the game. I've always viewed the rules as a way of settling disputes or uncertainty instead, so you start from the role playing and only resort to rules when you need fair adjudication or clarification on complicated situations. i.e. don't give me quotes from the rulebook, tell me what your character does and we'll work it out as part of the story.
When most of the games rules are about a thing that thing becomes the focal point. 5e also assumes pretty high amount of combat encounters per day to keep all classes in balance, if you are having less then it will make some classes just bad picks which can feel bad
Personally I don’t like it when people don’t play by the rules of the game we have decided to play together, so definitely things should work as the rules say and then ambiguous things are sorted with GMs world’s logic as “rulings”.
If you start by ignoring the rules and only consulting them when there is a dispute then I want to play another game with less rules to begin with