The issue of whether brands will be replaced by private labels is one that needs to be analysed with respect to different market and product segments. There is a huge difference between a prestige item like an Hermes handbag and a roll of paper towels.
Even during the recent global recession, high end luxury brands have had consistent growth, often over 10-15% annually. Part of what makes a luxury item worthwhile is the instantly recognizable prestige of the brand. When you buy Louis Vuitton or Cartier or Rolex, in part what you are buying is the brand name. This is not really something a private brand can replace.
In a different market segment, consider items like cleaning products and kitty litter. Brand prestige really isn't a crucial issue with these -- you aren't showing them off to your friends. Especially at the lower and mid-range price points, with items like these, people just want whatever is cheap and functional. For goods like these, private brands, due to lower costs, have an advantage.