Option 1: link chores to money.
The point of chores is that the child learns to spend a certain amount of time on regular maintenance, and they also learn the basic skills required to e.g. wash up, clean the house, keep tidy, whatever. They will need this when they grow up and move away. They are not doing them simply to make your life easier (though that is a nice side effect).
The trouble is that if you provide money as "payment" for chores you are implicitly setting a price of not doing the chore. So you need to decide what happens when the child decides to exercise the option. If you allow the child to skip the chore then you are losing the point of chores. If you don't allow it then you have exposed your "payment" as a lie.
Option 3: Provide money on application.
This teaches the child that money grows on trees, and the way to get it is to beg, wheedle and pester. They don't learn what money really is and how to handle it.
So I would very much recommend option 2: provide pocket money, and require chores, but don't link the two.
On "socialism" vs "capitalism", the family is not the place for such thinking, and the age at which chores and pocket money become appropriate is far too young to get into ideological questions.