Hobbes does make you think. In a capitalistic society, to do well means to outcompete others. So, yes, the natural state of war is still alive and well to some extent. On the other hand it's argued that wealth is actually created and no one loses. I'm not sure I buy this.
I don't see Hobbes advocating a society where no one has the right to own private property. He does say a sovereign may decide the property rights of his or her subjects. Perhaps property rights are used by the sovereign as a carrot to motivate his subjects to stay true to the contract. If we want to maintain possession of our possessions, we need to live in a state of peace and that is ensured only through continued willingness to adhere to the social contract with the commonwealth.