I feel kind of lost here. I hope someone is able to answer some questions (if not all, pick any).
1. What are all these "paths" for? I searched through this thread and word path is used for very different things. Would I be correct to say that paths are only Piety / Lewds / Knightly / Ambition and everything is else is "branching" as opposed to being a path? By branching I mean that you can pick how you solve a certain situation, there is only one branch and that's how you progress through the story?
2. The 4 paths that I mentioned, are they actually 4 separate paths and you have to pick one of the 4? What happens if you don't and you just keep going balanced between them? Or is Lewd supposed to be opposite of Piety and is Ambition supposed to be opposite of Knightly? If so, does this mean there you should pick one of each? Is there any locking out following these paths and would that mean that there are Lewd + Knightly, Lewd + Ambition, Piety + Knightly and Piety + Ambition "paths"?
3. What is the purpose of paths? Is it mainly about ending opportunities? If it's about locking away large portions of the game, why doing it in first place?
Is a good question, Being realistic, many stuff of the game doesn't make complete sense until some features are in place or are we there yet, and It's long overdue since I make the public post in patreon about Paths & branching explaining all of it.
I like put heavy branching in the game, but it has no sense to lock down big chunks of content, specially when people is playing a not completed game. So player have the option to experience most of it during a single gameplay, beyond each little branching. So no, "Stat Paths" Doesn't lock down the other "Stat Paths"
Path naming for some things could be misleading, as players started naming Paths to the branching, and myself I name the stats related quests "Stat Paths".
But yeah, Most of the game is branching: Either you pick A or B, you will end on C. Is different solutions to solve the same problem, as you said about branching.
Only linear thing in the game is main story content, that will mostly really diverge on the final stages of the game (Chapter 6), Let's say, the different endings.
Basically:
The four cardinal stats (official name) are not paths. They're four stats pool that can be increased, as you said, trying to get balanced, and the more points in each the better to unlock more dialogue options and little choices. Is hard to be the best in all Stats in a same playthru but it can be achieved (except in one exception that has no sense worrying about now).
There are at least one quest related to each path, and they're not incompatible between. They're like companion side quests (this will start making sense soon).
Only Final choices is what locks out other final choices. Like marrying choices, and the ending choices.
Let's say in the end of a "Stat Path" (A Stat related Quest) you can marry someone? Well, that for sure will lock marrying other people in the game. But that will not lock pursuing the other Stat related quests until you get to the point you cannot marry another woman. This is a made up example.
But all really blocking choices will only appear near the end of the game, so makes no sense worrying about it now.
game freez after i give the mayor the love potion
Have you used the Cheat button? As said in the changelog, game will bug out in some places if you use the Cheats.