Another way to "rebalancing" Lannister card draw without handing it out to everyone would be to lower the limit to 2, would make it less usefull to truly stack your deck with cardraw.
But in the end there are maybe too many problems with this changes
Martell already circumvents draw thanks to their reveal and put into your hand cards and they would become the new top carddrawer
For me the "biggest" thorn is
- Difficulty in building trait-themes decks
Although this seems to get resolved with the new deluxe expanions, can't wait to make a dothraki or clansmen deck (if those themes get chosen)