In situations like this, I have found the issue is torque. A jewelers screwdriver doesn’t really allow you enough grip to get decent torque to break the gunk-seal free. I typically will put the watch head in a vice (with cloth covering it to protect it) and put the jewelers screwdriver in a locking pliers. Then as long as the screw head and driver have a firm seat (not gonna strip), and you apply a bit of downward pressure over the top of the center point of the driver (I usually apply my thumb pushing down) you can apply enough torque to break it free....and hope you don’t snap the screw.