I realize the grips are shot, but I don't think I'd change them. They will get worse, and then they'll get even worse, which is too bad; but nothing else will look as good. Until they finally disintegrate entirely, that is.
Maybe you're right... The only issue I have is that the left grip is quite uncomfortable since the end is missing. The metal against my hand is no fun even on a short ride (I didn't put on gloves for its first outing), so surely a longer ride will be worse yet. It's a tough call for me because I wanna ride it instead of looking at it. Maybe wearing gloves will be a partial solution.
EDIT: A gentleman on Flickr informed me that the hub on the bike is a 1951 (the 1 on the hub is indicative of the year), so I believe this bike is a 1951 assuming the hub and bike were made the same year. I think that she's in great shape for a 60-year-old lady.