First, most of the land available was not acquired on a "first occupier" basis.
You're right, that is a problem.
Certainly none of the currently government-controlled lands were gained justly.
But is it a problem we can do anything about? In some cases, yes, in most cases, no. So the fairest thing we can do is to, in those cases where we cannot determine a rightful owner, simply call it a blank slate. Government lands go "up for grabs," and everything else is given to the current owner. By and large, everything in the private sector was acquired by voluntary trade, even if the original owner wasn't
actually the original owner. What's done is done, and we can't change the past. All we can really do is move forward, and hope to build a fairer future.
Secondly, and this is a problem (arguably) with both systems is that land can be over-claimed causing those who must make use of the land bound into serfdom to those who "own" the land.
I'm not sure what you mean by this. If you mean the oft-repeated criticism of the absent landlord, that's not really the problem you make it out to be. You can buy land from a landlord.
A third potential conflict is with those who have no need or want to "own" land but only to make use of it on an occasional basis (nomads, cattle drivers etc).
That's less of a problem in modern society, but there's no reason that a nomadic tribe couldn't claim their entire range, and a first-occupier rule would cover that. Just because you're not in all the rooms of your house all the time doesn't mean the rooms you're not in are not part of your house.