Problem is, with a non-contact tester you're only checking for HOT wires.
You're not checking for valid NEUTRAL wires. If neutral is working properly, it's cold.
If it's not working, it's also cold.
However you can trick the "dead part" of the neutral to be hot. Simply plug a lamp that is turned on into the dead socket. Wherever neutral is dead, it will now be hot (because it can't get back).
This may lead you to a faulty connection. Make sure to bone up on "how to make good connections" - avoid backstabs, use Wago lever-nuts, make wire nuts VERY tight, and torque lugs and screws to the specified torque don't guess.
You're not checking for valid NEUTRAL wires. If neutral is working properly, it's cold.
If it's not working, it's also cold.
However you can trick the "dead part" of the neutral to be hot. Simply plug a lamp that is turned on into the dead socket. Wherever neutral is dead, it will now be hot (because it can't get back).
This may lead you to a faulty connection. Make sure to bone up on "how to make good connections" - avoid backstabs, use Wago lever-nuts, make wire nuts VERY tight, and torque lugs and screws to the specified torque don't guess.