Get a Grip
Unless a fielder is very lucky, when they get the ball they will need to throw it somewhere. Thus, the throwing hand should be used to pick up the ball. It saves time and the glove is not really designed to make it easy to get the ball off the ground.
When the ball is rolling both glove and bare hand should be used together to pick up the ball in the bare hand. The glove serves as a wall to push the ball against while the bare hand is getting a good grip. This prevents the “off the finger tips” fumble of the pickup. Also, the hands can be moving the ball toward the throwing position even as the bare hand gets a better grip.
The ball must be picked up before it can be thrown. Until the fielder is ready to throw there is no need to see the runner(s). Therefore the ball should be watched while it is being picked up. There will be plenty of time while the fielder is moving into throwing position to look for runners.
In some rare cases the ball must be picked up and thrown underhanded in the same motion. This occurs when the ball and the play are close together (20 feet or less distance) and speed is the primary concern. In this case the ball must be “scooped” rather than “grasped” with the hand doing the scooping moving in the direction of the target. That way, if the ball is not secure in the bare hand the ball will at least be moving in the direction of the base and might get there in time even if it is bouncing.