New outlet doesn’t work breaker not tripped and the tester lights up
StackExchange will hear this form of a question many times, and even reading it right now it doesn't make sense. If you have a device that has gone bad and if the new one doesn't work. How does it not work? Either you replaced the receptacle correctly and you power is still off, or you installed the receptacle incorrectly now it doesn't work.
First you need to isolate the problem. You will probably need to abandon the tester light and get a voltmeter. Check the voltage between the source wire and the neutral and the ground wire. They both should read around 120V. If they do then the problem is the new receptacle. If they don't then your circuit is the problem. I would look real closely at the breaker panel and verify the breaker is not tripped. Just because it is not in the tripped position does not mean it hasn't tripped. Use your voltmeter to check the post on the breaker against the neutral bus and see if it is getting 120V.
If you have power at the panel and still no power at the circuit. Then you will need to trace out the entire circuit to find where the break is. Chances are you are not working on a dedicated circuit so find the other devices on that circuit and see if they are working. If they are working, try and find which device is feeding you receptacle and look to see if there is a loose wire in that box.
In short we isolate the problem and then look for the reason it isn't working.
Also most electrician will recommend not using the back stabs on a receptacle for installation. It is better to wrap the conductor around the screw and secure it that way.
Good luck and stay safe.