While entanglement assistance can increase achievable rates for classical point-to-point channels in the zero-error and one-shot setting, it comes as a surprise that entanglement does not provide any advantage in the asymptotic setting with vanishing error.
However, in the abstract of the Bennett paper which is cited, it says
Prior entanglement between sender and receiver, which exactly doubles the classical capacity of anoiseless quantum channel, can increase the classical capacity of some noisy quantum channels by an arbitrarily large constant factor depending on the channel
1) I thought that the noisy channel coding theorem states that in the asymptotic limit, one has either zero error (if the rate was smaller than the capacity) or arbitrarily large error (if the rate was larger than the capacity). What is the asymptotic limit with vanishingly small error?
2) Why does the increased classical capacity by a constant factor due to entanglement not change anything in this limit?
EDIT: While the answer given by Serwyn is an excellent and correct one, I had a misunderstanding in the question. Essentially, Bennett et. al are talking about quantum channels in the second quote whereas the first quote is about classical channels only.