I don't think anyone's really disputing that people can house rule anything they want. But, honestly, the OP kind of set himself up for complaints since it's pretty clear that class skills and saves aren't really meaningless. They help define the character of the class. The background is intended to give players a chance to inject additional, personal character concept aspects that aren't tied to the class.
Of course, I think the easiest way to handle it isn't to strip the classes of parts that define them - it's done through other choices. Want a better Dex save than a cleric typically gets? Put one of your higher stats in Dex. Take the Resilient feat for it (you not only get the save as a proficient one, you get +1 Dex). Want to add non-class-based skills? Take the Skilled feat. If the concept is that important, surely it can wait a few levels and take up a feat slot.