Don't we all have the right to a basic level of livelihood? Why should the family I was born to determine wether I have clean water to drink or not, for example?
Because of this thing called reality.
If the circumstances of your birth are unfortunate, that's sad, but it does not have to be other people's problem.
In the wild there is ownership (or possession), and animals defend it with brute force. Property rights regulate this and tend to enforce the concept of ownership in a civilized way.
Without property rights, we'd be back to defending ownership with violence, even on the individual level. Few people want that, and I'm not one of them.
You have the right to it, but you're not entitled to it.
To me, having the right to do something means you can't be prosecuted for doing it. It doesn't mean that society must help you.
As a practical matter, what do you think happens to any society that suffers a class divide severe enough that (a) the majority of the citizenry ends up in the underclass and (b) being in the underclass sucks badly enough to really piss people off?
Societies need to take care of their citizens and ensure they receive, if nothing else, fair treatment under law and approximately equal opportunity. Otherwise, you get behavior like this: https://youtu.be/HL45pVdsRvE?t=60
What a lot of people on both sides don't seem to be getting is that Trump getting elected was an expression of bipartisan anger and frustration on the part of voters. Their anger and frustration may have created an even worse result, depending on whether you think Trump is going to ease or worsen inequality in this country, but guess what? Stupid things are exactly the result of festering anger and resentment.
There are a bunch of folks out there -- maybe you're one of them, maybe not, I dunno -- who kind of wish for some kind of implosion of American society. They think they'll come out ahead, because they've got guns or they live in the middle of nowhere. But, they won't come out ahead, they'll lose. Everyone will lose.
So you can pontificate about the ideological merits of extreme Libertarianism all you want, but in practice it works exactly as well as extreme socialism.
It has a violent revolution. But your question contains an unstated assumption: that capitalism leads to "the majority of the citizenry [ending] up in the underclass". Feudal societies saw most people living in poverty. Communist societies saw most people living in poverty. Properly functioning capitalist societies have a healthy middle class, and a tolerable standard of living even for people at the bottom.
"So you can pontificate about the ideological merits of extreme Libertarianism all you want, but in practice it works exactly as well as extreme socialism."
The society that came closest to "extreme Libertarianism" was the United States in the 19th century. (Especially after the civil war). That was the century of the industrial revolution, huge advances in technology and wealth production, and an unprecedented rise in the standard of living.
I'm not sure what you mean by "approximately equal opportunity", since the circumstances of life are so different it seems delusional to me to hope for equal opportunities for everyone.
But I do agree on the concept of fair treatment under law, and I believe it's a fundamental basis for the notion of justice.
I just don't believe poverty necessarily results from any unfair treatment, and that as such its prevention or mitigation is not society's duty.
Let's take them one at a time.
1. Life - so one has the right to life, yet is not entitled to it? If I take your life and don't get prosecuted for it, then everything is cool?
2. Liberty - if your skin color was black in the Antebellum south, you had the right to libery, you just weren't be entitled to it?
3. Pursuit of Happiness - you have the right to say, be able to read, which I think most would agree is pretty essential to pursuing your happiest self. But you are not entitled to any basic education, and the onus should be on you?
This is one of the most cynical viewpoints I can think of.
Edits: premature eclickulation
Your third example is interesting because indeed, I personally believe that education should be a service like any other, one that society is not supposed to provide, at least not by law. On the other hand, law should not prevent your from getting one, nor should it allow anyone to prevent someone from having one. Hopefully you can see the difference.
It's like say owning a car. You have the right to own one. If someone steals it, it's a crime and society must punish it. But society does not have to provide you one if you don't have any.
You can argue that certain rights are fundamental, but it doesn’t mean anything if most people disagree with you. Whether or not rights are “fundamental” (i.e. exist outside of an enforcement structure) not a useful distinction, because it’s meaningless - whether or not you get to exercise that right has nothing to do with your claim being correct, it has to do with other people agreeing and sanctioning your behavior.
Like most normative claims, personal rights do not really exist coherently if you don’t first define them in the context of a relationship. This is demonstrably the case because the idea of a right would be redundant or meaningless if it were not for an adversary threatening that right in the first place.
