Is it unethical for the professor to continue to enforce the deadline and associated penalty (significantly reducing student A's grade and grade average) under these circumstances?
First, I think it's quite misguided to frame this as an ethics question. If the professor is creating course policies and enforcing them in good faith, with the goal of achieving certain educational outcomes, then, even in the event that the professor makes a mistake or does something that most reasonable people would disagree with, it would be hard to argue that the professor is behaving unethically. Thus, the correct thing to ask IMO is not "is it unethical" but rather "is it appropriate" or "is it the right thing to do".
With that said, I think the best thing that the student in your question can do is argue her case to the best of her abilities and allow the professor to make a decision to the best of his ability, and respect whatever that decision ends up being. Bringing ethics and accusations of unethical behavior into the debate is counterproductive in my opinion and may backfire by causing the professor to become defensive and entrench himself further into his punitive position. Personally I think that given the student's lack of timely action to communicate the nature of the problem to the professor, which resulted in missed deadlines, enforcing the course policy would be a reasonable decision. On the other hand, making an exception to take into account the allegedly extenuating circumstances would also be reasonable. To my mind it's simply not a black and white issue.
As for the student's supposed difficulty to communicate with the professor because she likes him... well, not sure what to say about that. People have all sorts of difficulties in life, but ultimately the question is, is the student willing to make an effort and take proactive measures to overcome her difficulties so as to achieve the outcome she wants to achieve? If so, there are any number of steps she can take. She can write a letter instead of speaking to the professor in person. Or she can recruit a friend to come with her to a meeting with the professor and help her overcome her shyness or even speak on her behalf. Etc etc. It's all a question of will. And if she lacks the will to confront the problem, she has all my sympathy, but nonetheless I don't see that as an "extenuating circumstance" and she should not be surprised if the outcome is not what she wished it to be. In any case, best of luck to her.