1999 GMC Jimmy driver door latch

Try the easy way first. Stick a screwdriver into the latch and then hold up the handle while pulling the latch open with the driver.
In your close up of the latch you would put the screwdriver in the left hand opening in the latch, Then while lifting the handle you would pull the screwdriver to the right.

What you're doing is making the screwdriver act like the striker itself and rotating the latch cam.

The probable failure in the latch is DIRT. They get dust and crud in them and the small spring that pulls the latch clip down sticks, Then it only takes a small tap to rotate the latch in.

Now in the event that you already tried that and the latch still didn't move the latch comes out without much trouble. Reach in the hole at the bottom of the door and find the latch. The clips on the rods just unlock from the rods by pushing them off the rod. Then the rod slips out of the clips. the latch then slides down the door to the bottom and out the hole.