If $1/day-wage-labor is wrong, then VOLUNTEER work is also wrong.
There is an obvious difference between volunteer work and paid work: paid work is done to maintain a household in some manner while unpaid volunteer work is not.
You can't dictate this difference between volunteer work and paid work. Just because this might be so in most cases, there's nothing preventing someone from doing paid work for some other need, such as saving for a later business investment, or because they want money for gambling, or they want money to donate to a worthy cause (or unworthy cause), or many other possible purposes. .....
"Maintaining a household" includes the maintaining (or attempting to maintain) the well-being of any household member which might include any of the possibilities in your . . .
So according to you, no job is legitimate unless it's done to maintain a "household member"? Which "household"? There are millions, or billions of households. So you mean THIS household rather than THAT one? i.e., the one the worker belongs to, and no other? So it's not legitimate for a worker to spend his earnings on anyone other than a "member" of the particular "household" s/he belongs to?
Or do you just extend "maintaining a household" to mean anything whatever is the worker's purpose, or anything to be gained, or in order to buy something for any purpose at all?
So "maintaining a household" is just a catch-all phrase meaning anything imaginable, as their purpose, that might motivate one to do that job. But if that's all it means, then even a volunteer worker is "maintaining a household" by doing work for free, by producing or gaining something from doing it and not needing to be paid money in order to gain whatever their purpose ("maintaining a household") is for doing that job.
Just because you come up with a term for it ("maintaining a household") doesn't mean they must be paid some wage you dictate as necessary but which is not their choice. If they choose to do it and be paid only $1/day or even no dollars at all, that doesn't somehow negate whatever their purpose is, or make one legitimate and the other illegitimate, just because you come up with some label to put on it. It's still their purpose, based on their choice, regardless that you name it "maintaining a household" or some other label you want to put on it.
If one worker is being paid $1/day and another worker is being paid zero dollars, both have their reason, regardless what name you want to call it -- either one is performing a task for their particular reason, and both are equally legitimate. You can't give any reason why one is permitted but the other must be made illegal. Just giving a name to something (e.g. "maintaining a household") doesn't make it legitimate or illegitimate.
Even if you can conjecture what someone's psychological motive is for doing something, in many cases, that doesn't entitle you to impose rules onto everyone doing something similar, who in some cases might not fit your psychological analysis, or conform to your set of rules defining their behavior, or presuming to restrict it.
If your rule was legitimate, then every employer must first test the job-seeker to make sure that "maintaining a household" is their motive, and reject anyone who might have a different motive than this.
Or, if "maintaining a household" is simply the label to put on ALL work of any kind, or any purpose one has for ever doing any work, then it has to also include volunteer work, which is done for some purpose which can be called "maintaining a household" (or whatever name you put on it) and yet not being paid dollars in return for it.
Whatever their purpose is, you're calling it "maintaining a household" which you can do if you want, but you can't dictate how much the worker has to be paid just because you call it this name.
. . . the well-being of any household member which might include any of the possibilities in your interminable wall of text.
Including volunteer work, which is one more possibility. Even my eternal infinite wall of text can be called "maintaining a household" if it serves some purpose I choose. Anything done for some kind of goal, however ambiguous, is "maintaining a household" (according to you) -- but you can't impose a certain minimum price on it which must be paid in order for it to be legitimate, and without which it has to be made illegal.
And to anyone familiar with economics understands, "well-being" is judged by the individual in question, not by an outsider.
That includes the price to be paid for it, judged by both the buyer and the seller, including when the dollar amount is low or zero in some cases (e.g. volunteer work) -- not to be judged by an outsider (anyone other than the individual buyer and seller).
Which means the obvious difference has been explained to you.
All you've explained clearly is that there is
no difference between volunteer work and paid work requiring one to be restricted but not the other. Both are done for the purpose of "maintaining a household" by your terminology, meaning simply whatever "well-being" is to be accomplished by the work, for whatever "household member" it might be who is to benefit. It could even be done for the benefit of an animal in the wild, or a tree, which is the "household member" -- anything which might be made better somewhere is the "household member" whose well-being is served.
The "household member" whose well-being is served is determined by the worker (seller), as all choices are that of the individual buyer or seller who judge this, never "an outsider."
Of course, whether or not you acknowledge that is a completely different question.
I acknowledge that you mean ALL work, paid or volunteer, is always for "maintaining a household" by your terminology, meaning whatever is to be gained from it.
And so there is no minimum price which can properly be imposed onto anyone choosing to buy or sell labor for their "maintaining a household" (whatever that might be), and so the price can be $1/minute or $1/hour or $1/day or zero dollars, according to their individual choice (buyer and seller) only and "not by an outsider" imposing it on them.
So I acknowledge your argument against minimum wage (or living wage) imposed by the state or other outsider, including that your argument is more succinct than mine.
And you're acknowledging that there is no legitimate purpose served by imposing a minimum price on the labor unless it's also imposed on the volunteer work, or -- there's no more reason to prohibit $1/hour labor than to prohibit volunteer work (zero/hour labor).