Um.

Considering her family's influence it's likely impossible.

I just wish she'd at least consider trying. It's not like they could make life much harder for her without starting to actually physically torture her.
And if she's scared of that, the chance that she'll die in the escape attempt would be a plus, cause they'd probably fall that low eventually even without any escape attempts.

Although I do agree with her at least trying to get away, you sometimes have to think about the time and situation that occurs.
From what I've personally seen from other isekai mangas, the noble families mostly live farther away from town. That is why they use carriages to get to places.
Also, there are not always kind people as the man who is willing to help her. If she walks out like how she is, then most people would believe she is a peasant, a servant, or a runaway slave. They would most likely bring her back to her parents, trying to get money from them for taking care of her.
Another thing, with the "dog's death" think about how many animals are left on the streets and you just happen to see one dead. That is what she thinks will happen to her: where no one will care enough to listen or help her and she would just die from undernourishment or starvation.

Once again, I do want her to also escape and find a strong backing, like the Monster Duchess and the Contract Princess, but she still does not know anything about the world outside. I would like her to fight for her right, I would also like her to turn the tide on her "family" by proving that her abilities are the best in their history, just like most other isekai.

Although you want a strong MC, you must also look at why the artist draws how it is shown in the manga. Sometimes the answer is by looking at other isekai with similar plot points.