No, this is not a "No True Scotsman" sort of argument or some sort of sarcastic joke; that's really what it means. It should stop bleeding in just a couple of days of regular flossing. If it doesn't, your mouth may be indicating another problem.
When I first started flossing I'd get some bleeding (in no small part because my lack of flossing before caused the gum at the bottom of the tooth to be inflamed). Bleeding is very infrequent now.
I personally use those plastic flosser sticks. I cannot figure out just the string. But I floss once a day every day and have for years.
Just remember to floss each tooth twice in a V shape. You need to do both sides of the tooth and the bottom.
I'm not a big flosser, so my gums always bleed a bit when I do it myself, but my dental hygienist always flosses me after a cleaning and she never makes the gums bleed.
Having said that, I love my waterpik. When I floss and rinse, it still flushes things from between my teeth that flossing couldn't get rid of. I highly recommend both!
There are also little dental picks you can get that you can use to clean between teeth without flossing, though I find them hard to use. If you're having trouble with regular flossing definitely ask your dentist for alternatives.
It's gentler than sticking a piece of plastic string between your teeth and more effective at getting trapped material and plaque out.
It’s not as smooth as others (but smoothness = less effective at removing plaque), so it may take some getting used to (esp if the tooth gaps are small).
https://en.wikipedia.org/wiki/Cetylpyridinium_chloride
Edits: spelling + link.