Your concern about your front axle is common.
If you have an open diffed front, and you HAD to pull the entire truck, it would mean the rear has zero traction...and if the one front tire slipped, you would still have zero traction, and slide down the hill....etc.
If the front is locked, you probably would STILL slide down the hill, because if going uphill, the rears carry most of the weigh due to the rearward weight transfer....so if the rear had zero, the front would not have more unless the front was on something different. On most uphills, the rear is carrying ~ 70% of the truck's weight.
If the front WAS on something different, say, with traction, and you had an open diff, and no tires slipped, the two front tires would be able to potentially hold you in position, or, potentially pull you over.
This would be the same if locked.
If the front had more traction that the rear (Again, it would need to be on something else than the better weighted rear...), it would typically have only about 30% of the truck's weight available TO pull.
So the fronts are almost never in a position where they are bearing the entire truck's weight, in practice, due to the rear weight transfer.
The fronts are typically used more like a hiker going up a steep trail uses his hands to grab a branch, etc, to help him up the hill. IE: His FEET might slip, but the hand gives enough purchase to hold position/gain enough boost to make it.
An open diff is like having hands that let go if the other hand slips.
A locker is like having two hands that can grab what they can.
If you are in a situation where you need to drag your self up a hill in FWD, that's usually an exercise in futility.
When I used to get a FWD company car once in a while, and I had to get up a steep icy driveway, etc...I'd turn around and BACK up the slope instead...to avoid that "drag".
I also love that diff video, its perfect for explaining how it works....I remember posting it several times over the years myself. Its the best one I'd seen so far, despite the corny acrobats, etc.