Should I pay my kid for chores or have them earn outside the house?

Both, and they teach different things, which is the part that usually gets missed.

The useful distinction is between work that is part of belonging to a household and work that is a transaction. Taking the trash out because you live here is the first kind. Cleaning a neighbor's gutters for twenty dollars is the second. A kid needs both and they should not be confused with each other.

Where families get tangled is paying for the first kind. Once every household task has a price, a kid can reasonably decline to help because the rate is not attractive, and something that was about being part of a family becomes a negotiation.

A structure that works for a lot of households: basic chores are unpaid and simply expected, and there is a separate list of bigger optional jobs that do pay. The car, the garage, the windows. That way earning is available without turning belonging into a wage.

What earning outside the house adds that chores cannot.

A price somebody else agreed to. Inside the house you set the rate out of love. A neighbor pays what the work is worth, which is the first honest signal a kid gets about value.

The possibility of no. Nobody says no to a kid doing chores. The asking, and the rejection, is most of what makes this worth doing.

A ceiling they control. Chores pay what you decide. A business pays more when they get better or find more customers, and that connection is the whole thing.

Practical version for most families: keep the household expectations unpaid, add paid optional jobs if you want a bridge, and let outside earning be where the real lessons live.