Only if they "claim" land before it's all become owned through someone mixing their labor with it one time and owning it indefinitely. If we started with a completely unclaimed planet and everyone could start spreading out and claiming land, that would be one thing, but how could we make it work when we're starting from the point of a tiny % of people owning most of the land already?

In anarcho-capitalist philosophy, it's generally understood that homesteading is how property is originally claimed, and that in a situation with finite land like we have on earth, eventually all or most of the land available will have already been homesteaded and become private property, and future generations of people will obtain land through barter, monetary transaction, bequeathment, gifting, etc. This is how it works (for the most part) in propertarian societies, even statist ones, but there are some exceptions in our present scenario. Namely, "publicly owned" land would be considered illegitimate and available for homesteading after the dissolution of the state.

I tend to agree with @kafkanarchy84 that such transfers of ownership should occur at a local level, where the federal government (if we are talking about publicly owned land in the states) simply releases ownership of said property and those who are already using it mixing their labor with it (most likely locals who hunt, fish, hike there, etc.) would be able to retroactively homestead it, while unused portions or places that were built specifically for government use like office buildings and federal courthouses would be available for new homesteading. In that scenario, perhaps the Rainbow community would be able to show retroactive homesteading of any lands that are used for gatherings.

There are other exceptions. Sometimes a piece of property goes unused or abandoned for so long that it becomes clear the owners (if they are still alive) do not care about it. In an instance like that, you might try contacting the owners and if none are found, go ahead and homestead the land. If someone turns up later who claims to have ownership rights to the land, the matter could be settled through arbitration. There have been at least two such properties that have entered my awareness in the past few years near where I live.

Related to this topic is your point from the article that:

At some point, there would be no "unused" land left, and everyone who didn't already "own" some would have a massive hurdle to overcome in ever breaking through this ownership-based classism.

As I pointed out above, this is already the way things work in post-raw land propertarian societies, and that has not resulted in people being unable to obtain land or in land ownership consolidating into the hands of a few.

There are several reasons for this. First, people change their minds. Sometimes we think we want to have an apartment in the city, and then later on we decide we'd rather have a little farm in the country. So we put the apartment up for sale and go looking for a farm to buy. Second, wealthy families tend to grow poorer as generations pass. (Tend to. There are of course exceptions to this, and most of them are royalty or in some other way state-connected.) This means that they end up having to sell their mansions, break up their country properties into smaller parcels, etc. And thirdly, ordinary people move up and down the economic ladder throughout the course of their lives. So we might start out renting and later become a landlord, then experience some kind of financial or health trouble and sell off our rental properties, and use the part of the proceeds to buy a one bedroom house for our personal use.

Real estate changes hands all the time in a capitalist society, and not only the wealthy have access to it. Many of the very poorest people on the planet count their privately owned land as their most important asset in life. It feeds them, shelters them, and is the only thing they will be able to pass onto their children. The greatest bane of their existence is likely to be property tax.

Furthermore, in free markets the hurdle to owning land for the first time is not "massive", as you put it, but just regular-old challenging, as is everything in life that is worth doing. If it were that difficult to become a real estate owner, there would be basically no real estate market.

EDIT: I forgot to address this question:

Is property gained illegitimately (through the state) removed from the current owners?

Yes. Absolutely. If the "owner" can be shown to have obtained the property through illegitimate means (paid for with tax money, obtained through eminent domain or conquest) the property should be returned to its rightful owners, or, if none can be located, made available for homesteading.