I thought this one would come up. No, the curses in the garden are not an expressed or implied mandate of a particular gendered male provider role. This is probably the biggest other foolishly misread Scripture on the alleged male provider gender role subject. I'm not sure which one is worse, the blatant word-swap that people do with 1 Timothy 5:8 (imagining that it says "any man" rather than what it actually says, "anyone") because this one, likewise, is a matter of applying really basic reading comprehension as well as a ton of other common-sense statements
Let's walk through the curses very carefully, applying actual logic. Hopefully I can get away with just summarizing these passages and will be received in good faith so I don't have to copy/paste large passages.
Okay, so here is what we see when God lists to Adam and Eve:
To Eve:
*multiplied pain in childbirth
*her husband rules over her
Now, let's see if I can really persuade people to use their heads in a really, really basic case of deductive logic: notice that God does NOT directly tell Eve that she will die -- makes no mention of it.
"To the man, He said":
*Work from thorns and thistles
Then with no shift in audience -- for example, it doesn't say "to both of them, he said" -- God continues to Adam that he will die.
This is an incredibly basic logical deduction. Again, let's do some REALLY basic a + b = c kind of deductive logic:
(a) Eve is not told that she will die
(b) we know that women DO die just like men do
(c) death is a declared as a consequence only after Adam is addressed, and after God tells him about thorns and thistles
a + b + c = God intended BOTH of Adam's curses as understood to apply to Eve as well as Adam, not just the second. It is not possible to read the text, without deliberate coercion, as God telling Adam that the first curse only applies to Adam while the second statement to him -- again, with no shift in audience -- applies to both, which it clearly does, unless someone wants to insanely argue that women never die since only Adam was told that he would die, not Eve -- it has to be both or neither based on a simple reading of the text.
Purely reading comprehension and deductive logic, given that women aren't immortal, make it plain in the first place: both curses stated to Adam also apply to Eve, not just the second curse (death). Instead people idiotically:
Part 1 (Thorns/thistles/hard work): claim this applies only to men (the "provider role").
Part 2 (Death): quietly, without explanation, allow this to apply to everyone.
I'm a white collar worker. I have never worked from the ground in my life -- not even once. I'm pretty sure that most of us understand the "thorns and thistles" curse as a metaphor for something that pretty much none of us can avoid because of efforts of working. Interpreting this as "making money" is the most baseless and arbitrary viewpoint possible. I think even most housewives would love to tell you that their work within the home is not devoid of its own forms of pain and frustration just like (say) a man's white collar job has his, as well as someone who literally works the ground, which has always included both men and women.
Either way, the "curse to Adam" defense of a male provider role is foolish beyond belief, whether within the confines of reading of the passage itself, or applying even the slightest common sense. The curse "to Adam," which is really to them both, implies no mandate to Adam that he is particularly tasked as provider any more than the woman is:
*1 Timothy 5:8 applies to both men and women
*"thorns and thistles" also applies both to men and women
Hence, there is no particular male provider role, and we're left with the principles that we all work, albeit in different ways, and family provides for family as they are able and as necessary.
You seem very sure of your logic in reading Genesis but I would ask you to go back and reread the passage.
The instruction given in Genesis 2 is where he told Adam NOT to eat of the tree and prescribed the punishment that would be wrought upon him if indeed he did eat of the tree.
Gen 2:16 And the LORD God commanded the man, saying, Of every tree of the garden thou mayest freely eat:
Gen 2:17 But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.
Now, to the punishment put upon Adam and Eve...
The curse on Eve is hers alone to bear.
The curse on Adam is his alone to bear. What was the curse put on Adam? That he would die? No, that had already been prescribed.
In Genesis 3 he is not informing Adam that he will die, he had already informed him of that in the instruction not to eat of the tree.
In Genesis 3, he is simply telling Adam that until the day that he does die, he will eat in sorrow out of the ground all the days of his life.
He repeats this in verse 19 by saying "In sweat of thy face shall thou eat bread, till thou return unto the ground;"
Gen 3:17 And unto Adam he said, Because thou hast hearkened unto the voice of thy wife, and hast eaten of the tree, of which I commanded thee, saying, Thou shalt not eat of it: cursed is the ground for thy sake; in sorrow shalt thou eat of it all the days of thy life;
Gen 3:18 Thorns also and thistles shall it bring forth to thee; and thou shalt eat the herb of the field;
Gen 3:19 In the sweat of thy face shalt thou eat bread, till thou return unto the ground; for out of it wast thou taken: for dust thou art, and unto dust shalt thou return.
So, again, this is spoken directly to Adam and not to Eve. The overall punishment for eating of the tree is death in the day that they eat of it. That punishment is not being declared here except as a way of saying what Adam will have to suffer until the completion of that punishment.
It does clarify that he will return to dust but that is not a new punishment being given, it is simply an explanation of how the punishment will be done.
