An alternate alignment system isn't such a terrible idea in this regard.
In fact, something like this certainly gets more at the heart of "who are you and what drives you?" than the classic alignment system does. "good" and "evil" are so hopelessly nebulous and "law" and "chaos" can be entirely unhelpful too as it often depends very much on where the rules on coming from in regards to a character's likelihood to obey them. And someone saying that they follow their own internal code is well... totally not helpful, everyone follows their own internal code.
I think that's it right there.
I feel like traditional alignment fits in campaigns in which law and chaos, good and evil, really mean something and are thematic to the events. But often alignment gets ignored because it's just not important to the storytelling.
I guess this whole experiment shows me that you CAN use alignment as part of the world building.
In fact, it would be fun to develop unique alignment axes with the players as part of a Session 0!
For example, if the group decides they want a campaign themed around survival and the outdoors, you could ask:
Who are you?
Woodsmen, travelers, barbarians, druids...
What do you want?
Conservation, hunting, exploration...
So you might wind up with:
Cosmopolitan - Rural - Wild
Conservationist - Explorer - Poacher
Or something like that!