Not only must there be alignment, it must also result in appropriate rewards and punishments in-game. For example, if a society is defined as being generally good and especially lawful, where stealing is considered a crime, then if in-game a player resorts to stealing from people in town and they get caught doing it, or even just bring suspicion upon themselves, there ought to be in-game negative consequences. I utterly disagree with and reject the notion that this amounts to "punishing a player for playing the game their way." Same thing if you have gained a reputation as someone who slaughters people for your material benefit or just for the fun of it. The excuse that 'I want to play the game my way' should not be allowed to dictate the game responding appropriately to such behavior by the player based on the values of the society within which the game is set.
I want to invite you to consider that You are conflating Alignment and reputation. Alignment is an internal barometer of worldview that determines your ultimate place in the wheel cosmology after death. It can reflect in your actions and it can also change with experiences.
There has been a bad trend (Pathfinder) of making actions reflect on changes in alignment, like its some sort of constantly updated actual moral Barometer. It also broadcasts that alignment which is another bad take.
5E - thankfully - has taken the opposite tack - Alignment is not broadcast and its just an internal point of view that can change with experiences and exposure to different things - less actions you take and more based on experiences.
Reputation is a totally separate mechanic that should work as you describe. However, a Chaotic, Neutral or Lawful Evil person can have a good reputation - especially if it suits their needs in the long or short term.
