So i guess you used the Uno schematics to find out about that lmv358, right ?
This part is quite small, probably too small for you to get your measuring pins to it.
As you can see, SCK (pin 19 of the ATmega, and pin 13 on the socket next to the LED) control that lmv358.
You can do some measurements at pin 13 to see whether that does what you expect it to do.
If you press the reset button, does the LED stay off too ?
If it blinks a few times upon reset, then your sketch uses pin 13 and keeps the LED off.
In that case there's nothing wrong.
You can also try the blink sketch, which does nothing but set pin 13 high and low in a relatively slow pace.
If there is no blinking of that LED, does an other LED (with the accompanying resistor) on pin 13 blink ?
So multiple ways to check and see what's going on.