Removal of property rights intrinsically takes people's ability to provide for themselves away and replaces it with dependency. Some people seem to prefer this arrangement and I can respect that, but when they describe it as 'liberty', I assume that they must have lost their minds.
That is a direct parallel to the entire point of the "property is theft" argument: that strict enforcement of property rights is a limitation on peoples ability to make use of property, by monopolising use of it.
You can destroy the ability to make use of property by going too far either way.
A good example of how carveouts can increase the ability of people to make use of property and improve their ability to provide for themselves is the Scandinavian "freedom to roam": In Norway anyone can use "outland" - areas that are not part of urban settlement. Owning the land gives you a monopoly on certain users (e.g. you will own hunting rights and forestry rights, for example) but far fewer than most other places.
Apart from providing liberty in the form of being able to go where we want without having to ask permission of land owners, for centuries this has been an important means for people to help provide for themselves: Nobody can deny you the ability to e.g. pick berries or mushrooms, or let your sheep graze, for example. It provides a substantial increase in utility of land that would often see substantially less use, and makes a lot of people able to be independent and work for themselves instead of someone else. That is one way it can increase liberty.
Defining liberty as an absence of dependency cannot work in any society because it's going to degenerate into an unfree one more or less immediately. What's worthwhile to pursue is that all rules apply to everyone in the same way and that everyone has a realistic opportunity to shape the rules together with all the other people. This means participation in the political process, division of powers, transparency to avoid corruption, education for everyone, and enough executive power to prevent some random billionaire from unilaterally seizing power. It does not mean abolishing the organization tasked with guaranteeing those rights.
Not related to what I said.
EDIT: We even have good historical examples of the latter, such as the Great Famine in Ireland, where food exports continued - people died because other peoples property rights were put before their right to life, and people had no realistic prospect of acquiring or growing food in time to compensate for the crop failures. Notably, in contrast, in at least one past famine, the government had banned food exports to force prices down.