Let us discuss how you should understand the parts that are problematic.
You have written:
"But when I am reading the abridged version (God knows what will happen if I read the unabridged version), the text is so hyperfocused on one's family and ancestors it feels obsessive. On almost every single page, there is one reference about how one should do this or that just because he/she is a kshatriya/brahman, or because he/she is from 'Bharata' race or because he/she is the child of this or that god and so on. Krishna, when convincing Yudhisthira to fight Jarasandha, says that he should do so because of his 'noble birth and illustrous race'. This level of dependency and judgement based on one's family feels very immoral. No wonder why varna system of Vedas became corrupted to caste system in modern India. My own ancestors have done many wrongs. Does Hinduism tell me to follow my grandparents and do those wrongs? Does Hinduism tell the society to hate me because my ancestors did wrongs? Recently, one of the children of Hamas became ex-Muslim and is exposing Hamas. Will Hinduism shun him because his father, grandfather, and pretty much all his family is a bunch of terrorists?"
Mahabharata is great art and would contain elements that would appear strange to a modern reader. Would the characters speaking in modern style be believable to you? They wouldn't be. You have to read through the style and pick the gems. You have to use judgement in order to read such an ancient text. You are not supposed to accept everything in Hindu scripture. Only take those teachings that make sense to you, the essential parts.
What are the essential parts? How would I know the essential parts?
The goals of life, the purusharthas, are dharma (righteousness), artha (wealth), kama (desire) and moksha (liberation from Samsara). Mahabharata itself claims that teaching about these goals is its aim.
The wisdom of this work, like unto an instrument of applying
collyrium, hath opened the eyes of the inquisitive world blinded by
the darkness of ignorance. As the sun dispelleth the darkness, so doth
the Bharata by its discourses on religion, profit, pleasure and final
release, dispel the ignorance of men. As the full-moon by its mild
light expandeth the buds of the water-lily, so this Purana, by
exposing the light of the Sruti hath expanded the human intellect. By
the lamp of history, which destroyeth the darkness of ignorance, the
whole mansion of nature is properly and completely illuminated.
Mahabharata Adi Parva Section I
You need the first three goals to live as a householder. Does the flowery talk of race, noble birth etc. help you live a good pious life as a householder? If the answer is no then just ignore them. Treat them as simply old style of language. Only accept those parts that enable you to live a good householder life.
Examples of an essential part that will be helpful to a householder
Yudhishthira's views on Dharma
Yudhishthira said,".. I never act, solicitous of the fruits of my
actions. I give away, because it is my duty to give; I sacrifice
because it is my duty to sacrifice! ..I act virtuously not from the
desire of reaping the fruits of virtue, but of not transgressing the
ordinances of the Veda, and beholding also the conduct of the good and
wise! My heart.... is naturally attracted towards virtue. The man who
wisheth to reap the fruits of virtue is a trader of virtue. His nature
is mean and he should never be counted amongst the virtuous. Nor doth
he ever obtain the fruits of his virtue! ... if the virtues that are
practiced by the virtuous had no fruits, this universe then would be
enveloped in infamous darkness. No one then would pursue salvation, no
one would seek to acquire knowledge not even wealth, but men would
live like beasts. If ascetism, the austerities of celibate life,
sacrifices, study of the Vedas, charity, honesty, these all were
fruitless, men would not have practiced virtue generation after
(Mahabharata, Aranya Parva XXXI)
Bhishma on Dharma
Bhishma said, ‘..it is difficult to say what righteousness is. It is
not easy to indicate it. No one in discoursing upon righteousness, can
indicate it accurately. Righteousness was declared (by Brahman) for
the advancement and growth of all creatures. Therefore, that which
leads to advancement and growth in righteousness. Righteousness was
declared for restraining creatures from injuring one another.
Therefore, that is Righteousness which prevents injury to creatures.
Righteousness (Dharma) is so called because it upholds all creatures.
In fact all creatures are upheld by righteousness. Therefore, that is
righteousness which is capable of upholding all creatures. Some say
that righteousness consists in what has been upheld in the Srutis.
Others do not agree to this. I would not censure them that say so.
Everything, again, has not been laid down in the Srutis.’
[This refers to the well-known definition of Dharma ascribed to Vasishtha, viz, “That which is laid down in the Srutis and Smritis is Dharma.” The defect of this definition is that the Srutis and Smritis do not include every duty. Hence Vaisishtha was obliged to add that where these are silent, the examples and practices of the good ought to be the guides of men, etc.]
Mahabharata, Santi Parva, Section CIX
You need the last goal, moksha, to escape from the suffering of Samsara. You need to read the Bhagavad Gita to know how to attain moksha. I will give an example of why one needs moksha.
Yudhisthira said,’… Even thus, endued with actions, creatures come
into this wheel of life that is continually turning like the wheel of
a car, and even thus, coming thither, they meet with their
fellow-creatures. He, however, who abandons the worldly course of
life, which is really a fleeting illusion although it looks eternal,
and which is afflicted by birth, death, decrepitude, disease, and
pain, is sure to obtain happiness. When again, the very gods fall down
from heaven and great Rishis from their respective positions of
eminence who, that is acquainted with truths of causes (and effects)
would wish to have even heavenly prosperity? ..... Reflecting on these
circumstances, this nectar of wisdom hath come to me. Having attained
it, I desire to get a permanent, eternal, and unchangeable place (for
myself). Always (conducting myself) with such wisdom and acting in
this way, I shall, by betaking myself to that fearless path of life,
terminate this physical frame that is subject to birth, death,
decrepitude, disease, and pain.
Mahabharata Santi Parva Section IX
Word of Advice
You may accept or reject this advice. An ancient text like Mahabharata is like a sea bed where there are pearls hidden. You have to dive below the sea to reach the bed and find those pearls. The story is simply a frame work to teach the reader how to live.
It is a bad idea to read abridged Mahabharata. You are depending on someone else's idea of what is important to the reader. There is no substitute to reading the entire 4500 page text. The task is onerous. You will find portions of text that you will disagree with. That is perfectly fine. Just take those parts that in your judgement will help your spiritual growth.
It would have been better if you started with an introductory text.