I'm gonna take a stab at trying to answer this.
Some may have alluded to parts of this in the comments to the question however I'm gonna attempt to pull it together in a way that adds up for me.
Basically, history up until Avraham can be viewed as a failed attempt of natural morality to produce a just and moral society.
More was necessary to keep the world on track.
A core group of extraordinarily God aware people became necessary—in tandem with the general moral code—to keep awareness of God front and center.
A ממלכת כהנים וגוי קדוש to inspire and guide the world.
No reason is given for Avraham's selection.
Someone was needed to be chosen to head up this new core.
It could have been anyone that would prove to be a good choice, which Avraham was certainly shown to be.
His family blossomed into a nation and eventually received this additional Torah, which they agreed to.
In this reading, for humanity's inborn compass of right and wrong to function properly it became necessary to have a core of ever more God aware folks than the rest.
It is well nigh possible that born Jews never personally accepted the extra-moral parts of the Torah and yet God commands us in them and holds us to account.
Everyone is born with different life circumstances, be that as it relates to gender, health, smarts, looks, talents and ones family situation to name a few.
Everyone contributes to the smooth running of society in their own way.
I'm not getting involved with the potential randomness or arbitrariness of it.
Being that a group such as the Jews is a necessary ingredient in the preservation of society someone's got to do the job.
The fact that the world has shown that it requires such a group to keep it on the straight and narrow morally speaking makes it just for some people to be pressed into the job, especially as their forebears have already agreed to it and they've ostensibly been raised and trained to conform to it.
If a Jew is unaware of his heritage then he obviously will not be taken to task for the additional commandments however if he is aware of them then he will be, despite him not necessarily being amenable to them, via the coercion of punishment.
The world needs different people to make it work and everyone has to do their part.
That someone is born into more demanding life circumstances is no different than say someone born with any other type of difficulty or impediment that needs to be surmounted.
I'm not dealing with the potential lack of fairness that presents itself in anyones individual life circumstance.