As requested, PICTURES!
With everything removed this is what I was looking at (the axles are pushed in as far as they go):
A little distance on that picture:
Hub end of the axles: No visible obstructions.
All that gear oil! Brakes are COVERED!
And the side that the seal wasn't compromised:
Everything is all buttoned back up now. Might just bring it to a shop. Sucks working in the cold on the ground covered in gear oil. I'm still very curious why I couldn't get them to slide in.
As you all can see, the pinion pin is removed and in a box on the ground. The top two pictures, as I stated, are of the axles pushed all the way in! Both axles do not go in further than that. I used a 2.5lb sledge and some wood to pound both of them in, no luck. The c-clips spin around the axle shaft, it isn't like they're seized to the gears or anything. Both axles move back and forth about 1/8". On the outboard side, I cleaned the axles up, thinking maybe it was corrosion or something on the axle keeping it in (I saw BartonMD posted something similar on the OS when doing his S10). No luck.
I pin the pin back in a few times and changed the orientation of the carrier (I think it's called, the big thing that holds all the little gears in) by turning the drive shaft, also no luck.
It really feels like there is something holding them in. I've done a decent amount of gearbox work on windlasses on yachts, and generally when things are well oiled and free-moving but don't come out, it isn't a matter of a bigger hammer, it's a matter of removing something else.
I was tempted to start taking out the spider gears, but I decided that that wouldn't help anyways, since the two vertical gears are held in by the axles and c clips. And from what I've heard about resetting them, it sounds a bit like playing 52 pick-up.
I'm starting to think that maybe the bearings have created a channel in the axles. I don't think they're bad enough for that, though. Very little noise - it's the seal that I thought was the issue.