Girlfriend (22F) convinced herself that I (23M) gave her a bull**** ultimatum, I'm at my wits end with her.

We have been together since December 2020. I haven't met her family due to her Mother being paranoid of getting CoVid.
Back in March 2 major things had happened. She had a family friend who died and she was in grief that week (This will matter later)and we had a discussion that really put me off. She is Muslim and I'm Christian. From what she has told me I need her family's approval for marriage, from her Brother specifically. When I was told this information by her I asked her:
Me: So if your brother does not approve of me you would just break it off with me there?
Her: I mean yeah, what's the issue?
Me: So if your family just straight up does not approve of me because just simply just don't like me you would just end it with me?
Her: Honestly yes because I only have my mother and brothers and my family is extremely important to me.
At that point I felt like I was just waiting around to just be told no. The reason I had an issue with this is because:

1. She was basically letting her brother dictate her marriage and happiness.
2. Her Mother has expressed what kind of ideal man she would like for her, I don't fit that description.
3. I told her that our relationship should never have a 3rd party involved and that's exactly what is going to happen.

I had went home after that and by next day she was saying sorry and basically telling me that even if they said no that she would fight for me and she doesn't want this to end on a bad note. We talked it out and stayed together.
Fast forward to late August, we going through a rough patch right now. She and one of her brothers are staying at their friends Mothers house kind of to keep her company for the week. I get off of work and she says she wanted to call about something. Alright cool that's fine. She was basically asking me about how I was about to leave her back in March over her choosing her family over me. She then tells me how I always express that I love my sister to death which is true, and proceeds to ask me if I would choose my sister over her if my sister did not approve of her. I replied with:
"No I would not, because my Sister or even my Mom do not dictate who I should and should not be with. If they had brought up how they don't like you? Ok cool, they can feel that way, If they had pointed out red flags like how you had no ambition or were just disrespect for example then I would take that with consideration. Ultimately it would be up to me, but all they have are opinions just like any outside party should."
She then tries to say that I would putting a lot of pressure on her because it happened to be the same week her friend died, basically victimizing herself into thinking I was wrong to bring that topic up on the week her friend died. I didn't confront her on this because she would of made it into something it never will be.
She then tells me that I have a lot of family and that she only has her Mothers and Brothers so it is easy for me to cut off family like it nothing. While I do have a lot of family, I don't speak or reach out to a big majority of them so that was not really a good excuse for her.
Lastly the part that pisses me off the most is that I initially told her that I did not come here to separate her away from her family. She says that is what is feels like because of how our conversation went in March. And when I tell her that no person would be comfortable being in that position no matter how you try to justify it. And now I'm on the verge of just breaking this all off.
The problem is she is letting people who contribute to her misery dictate her happiness. And to be fair she can be a pushover when it comes to her family from the way she vents to me about them.
Can I please get some thoughts and advice on all of this? I know its a bit scattered and sorry for that. I am just very, very annoyed right now.
TL;DR: Girlfriend convinced herself of an ultimatum I never gave her and believes I took advantage of her grief as a way to manipulate her.
EDIT: There has been some solid advice and I thank everyone who did. Also.....to clarify because some comments are getting it twisted. I NEVER gave her an ultimatum and no, I am NOT telling her to pick her family over me. I can only speak for so many people but basically no one's relationship should be dictated by an outside party. Some of you see that as fine, I see it as a problem. Religion or not. I have no issue with being part of the family. I have a problem with family being too involved in a relationship/marriage.


Source.