power to outlet until something is plugged in
Assuming the breaker doesn’t trip and there isn’t a GFCI involved, this would generally indicate a poor, high resistance, connection somewhere in the line. Your voltmeter pulls so little current that you see the full voltage but when you connect a real load, the poor connection drops the voltage. 

Since you replaced the breaker and outlet, the problem must be elsewhere. Wires generally don’t just fail, unless someone drove a nail through it. So you are looking for a junction box somewhere with splices. At least hopefully, the person who wires it followed code and didn’t just splice it in a wall.