Why isn't liver bitter? I know that the liver is the organ that produces bile. I also know that it has a lot of bile ducts leading to a hepatic duct. Given that all of this bile duct and bile is in liver and bile is bitter(just from my knowledge(Mainly from when I have had a stomach virus)) I would think that liver is bitter.

However, I had home cooked beef liver once with onions and it was delicious. It wasn't bitter at all and it was tender. It was like chicken when it comes to tenderness. The taste wasn't like chicken or beef but nevertheless it was good.

Now this isn't a fresh liver(so not like the liver you find in whole chicken when it is sold) so that might account for part of it but still, with all this bile duct and bile that is in liver(even without a single piece of gallbladder), why isn't liver bitter from it?

I mean when I have a stomach virus, I know what I throw up via taste(Acidic and generally yellow(enough to cause me to have heartburn for a few minutes afterward), stomach acid, bitter and generally green, bile, any color and generally tastes like what went in + acid, food, nothing but water, throwing up unsuccessful, saliva). 

So if bile is bitter and liver contains bile, then why isn't liver bitter? Is the bile cooked out or something?
A few items that when added up result is a lack of bitterness:


Though bile is produced in the liver, it ends up being stored in the gall bladder for both humans and cows. I wouldn't expect the liver to be full of bile as a result.
One of the main functions of the gall bladder is to concentrate the bile by removing water and electrolytes from and making it, presumably, more bitter in the process. This also makes liver bile more soluble and thus can dissolve into water or other common cooking liquids more easily.
Typical butchering of the liver removes most of the hepatic duct, which is where a lot of the bile in the liver would be.