First, why would you want social egalitarianism, presumably because you are "a strong believer that people should be treated with respect"? So what does treating people with respect entail?
I'd argue that respect means primarily respect for their autonomy, that, as Kant put it, they are an ends in their own right, not just a means to your ends. That pretty much seems to guide all of the remaining ethical choices we have to make. In addition to Kant, I would also argue we need a notion of William Ross's prima facae virtues, to escape from some of the more tangled webs that Kant's ethics alone can get us into.

However you go about it though, it seems to come back to the fact that we should not exploit others just because we can. We are a social species and we rise, or fall, as a group, not individuals. This is supported by evolutionary theory, game theory, most religions... whatever you pick, the idea of co-operating with, rather than exploiting, others is central to ethical behaviour.

So, take your assertion that "Everyone should have equal access to shelter, food, water, etc/ an equal shot at life", that would seem to follow directly from our central ethic, no matter how we arrived at it. I shouldn't take someone else's shelter/food supply/water supply, just because I can because it is obvious that a person would want those things and I should not exploit their weakness at the expense of their ability to pursue their own ends.

But this has already happened and the problem we need to address now is how to undo it. The earth did not come into reality neatly divided up into parcels, one for each person. Every scrap of land that anyone owns was once taken from someone by force, the weaker party's shelter, food supply and water supply was stolen from them, which we've just established was immoral.

Now we might not find it moral to ask the ancestors of these people to correct the mistakes of their predecessors, but we'd have a hard time suggesting that their profiting from it is a moral act.

Your "...it is up to that person to take that opportunity" morality is tied intrinsically with the first half, that such opportunities are equal, but they are not. What constitutes the difference between an 'opportunity' and the instruction of a tyrant? If I were to order you to scrub my floor on pain of death, why would that not be an 'opportunity' for you to avoid dying? From a Kantian perspective, it would be because I was treating you merely as a means to my ends and not recognising your autonomy to decide your own ends.

So this is where the problem with denying the Basic Wage principle comes in. We do not all have equal access to land, food, water in any manner outside of the will of some people. Therefore, anything we must do to obtain our shelter, food, and water has us acting as a means to the ends of those people, not and end in our own right and is therefore immoral.

If each person had free access to that which facilitates life, without the constraints imposed by others trying to use us as means to their ends, then it would be reasonable to argue that the lazy would deserve their own suffering because they had chosen to be idle instead of doing the work which was necessary to secure their survival. But that is not the world in which we live. The people currently holding the resources we need (whether that's employers or potential customers) get to dictate what tasks we must undertake in order to obtain them, they are using us solely as a means to their ends.

A basic wage would mean that our own ends could be genuinely that, any contract we entered into would be solely because we have decided that it serves our own ends in some non-obligatory way and the other party entered into the same contract for the same reason, neither are treating the other solely as a means to their ends because the basic requirements of survival have already been met.

I'm just curious to hear people's thoughts on the matter. — Intrigued

I believe that all people are created equal and are endowed with certain unalienable rights. That among these are life, liberty, and the pursuit of happiness. So, people are equal because they have these rights. I don't think that's too different from what you said, but I'll take it further.

I don't mind that many people make more money than I do and I don't feel guilty that even more make less. I will say this though - when inequality of wealth and income becomes too high, especially when many people don't have enough money to live decent lives, there is something wrong with the society. Is that a call for fiscal equality? What about personal responsibility? I guess I don't care. I'm a pragmatist. We should do what's best to solve the problem. It seems convincing to me that a guaranteed minimum income may be a good method for moving in that direction.

You might be interested to know that conservative economists (like Milton Friedman of the University of Chicago) once proposed a minimum guaranteed income, whether you did anything to earn it or not. It was free money given to people on a silver platter. (The guaranteed income was a good deal lower than $40,000 a year.)

Why would anyone, especially a conservative economist, propose such a thing?

1. It would be cheaper to administer than the various benefit programs like welfare or unemployment insurance.
2. It would facilitate social and vocational mobility. People could afford to quit a job they disliked and search for a better job, without fear of starvation and homelessness.
3. It would be good for the economy, because people receiving a minimum income would probably spend all of it on basic goods and services.
4. It would result in greater equality of income (the rich would probably have to be taken more heavily than they were at the time to pay for the program.
5. It would be easier for people to return to school for more training.

and so on.

Marx did not propose rigid egalitarianism: He said, "From each according to his abilities; to each according to his needs." Some people are capable of producing more than others, and some people can produce less. Some people need more material goods than other people. (for instance, parents with three children need more than parents with 1 child or people with no children.)

Perfect egalitarianism is very utopian, Humans did not evolve to produce utopias, and are especially not enthusiastic about living in most other people's utopias. Perfect equality, perfect happiness, perfect contentment, etc. lead to hells of rigid control. How can you have perfect equality without perfect control of what people do? I don't want to be that tightly controlled.