Why are X-linked illnesses less common in females if females have X-chromosome inactivation anyway?
Barr bodies (X-chromosome inactivation) don't form in the initial fertilized embryo — it's not that one X-chromosome is inactivated, and then that same inactivation is passed down to daughter cells. Rather, X-chromosome inactivation occurs on a cell-by-cell basis in differentiated cells.  Note how the accepted answer to the question you linked mentions that different cell lineages will have different X-chromosome inactivation patterns — in other words, inactivation occurs later in the differentiation and proliferation processes than I think you might be assuming. 

To use your example: the liver cells do not all derive from a pluripotent (undifferentiated) cell with one inactivated X-chromosome, but instead derive from said pluripotent cell, and then undergo X-chromosome inactivation.  

Although the jury is out, scientifically speaking, on the mechanisms of selection of the X-chromosome to be inactivated, it's been postulated that some sort of decision-making may occur on this cellular level, resulting in a higher likelihood of inactivating the damaged or deleterious chromosome, if one is present. 

Regardless, as the accepted answer to the question you linked mentions, the 50% of functional cells that would result from chance or random X-chromosome inactivation tends to be sufficient for the body's needs.