What is the best way to get rid of the ink-out line on clothes labels?

Try spraying it with hairspray and blotting with a tissue. The lines are usually permanent ink, though, just to keep people from removing them. ; )

Why not carefully cut the label off?

Try alcohol, I use it to remove ink from clothes.

Alcohol is the solvent for most ink. If it's something like a Levis label on the back of jeans you may lose some of the print along with the ink.

Hairspray (contains alcohol) or just alcohol.

This can be tricky for a couple reasons. You might be able to get the ink off the label by soaking in rubbing alcohol and blotting it off if it is ball point ink, but you also run the risk of transferring it to your clothing item, so you'd need a way to isolate the label from the item. If it is permanent marker there are some stain removers that say they will get rid of it, but again, you may damage the clothing item along the way. My suggestion would be to remove the label entirely...and if you just have to have it for prestige sake, remove it, try the stain remover and resew it back into the garment.