What do you think is in that juice? Sure the cattle are bled after slaughter, but you never get all the blood out, even after it's been hung, which is still retained in the tissue.
EDIT: Ah ha! My apologies and I stand corrected :) Apparently the red colouring in the steak juice is myoglobin and not in fact blood:
https://cooking.stackexchange.com/a/8964/33
https://cooking.stackexchange.com/a/32619/33
http://www.exploratorium.edu/cooking/meat/INT-what-meat-colo...
https://www.thedailymeal.com/eat/stop-calling-juicy-steak-bl...
So you learn something new every day.
I get the general argument being made, but if this were true steak tartare would not be a thing.
Burgers served medium-rare should (probably) not be made from beef you buy ground.
Why would that be the case?
The reason we can cook and eat steaks rare is because dangerous bacteria form on the outside surface of the steak, and cooking the outside while leaving the inside rare destroys the bacteria. Grinding the beef mixes the surface of the beef with the inside, meaning you should cook the inside to 165F as well.
That's my understanding at least...
Ground meat on the other hand can introduce those harmful surface microbes to the rest of the meat, so burgers have to be cooked all the way through to ensure those microbes are successfully deaded.
The blood might cook out of the final burger, but it affects the taste and texture. In that respect it's not unlike cooking with wine, liquor, or cider, with which the alcohol will cook out.