The correct way to fit strings in a pin bridge is to gently push the ball-end into the pin hole using the pin, keeping a little tension on the string.

When you feel the pin slip past the ball-end, push the pin the rest of the way but without any extra pressure. You will find the string will not pull the pin out, because the pin is doing its job - to deflect the ball-end so that it is pulling against the undeside of the belly, not the pin.

You should also find that the next time you change the strings, the pins should come out easily without the need for a tool. Simply push the string down the hole a little, and the pin should come out without any effort.

My Yamaha FG260 (circa 1970) has been at concert pitch for its entire life, and is showing no signs of fatigue.