To not feel bad others have accomplished more?
No; I think the self-made man is dangerous because it leads people to believe 2 things:
1.) You can and should go it alone
2.) Other people shouldn't need your help.
I don't think Arnold tells his story the way he does in order to diminish his own accomplishments. It's used to highlight all the people that helped him along the way to get him to where he is. It's important to ask for help, to ask for guidance and depend on others. There's no glory in going it alone and you can be propelled immensely by relying on others to support your efforts.
Furthermore, it's important to provide a lending hand to others. Great men are built with the help of others and you have a responsibility to help others excel when you can. I think telling these stories help break the hyper-individualistic ethos of the American self-made man. Once you break this myth that so-and-so is self-made, we can make steps into re-examining how we treat our families, friends and fellow Americans (and/or countrymen). Common adages such as "pull yourself up by your bootstraps" will become more and more asinine once we understand no one is fully self-made; and I think that change in social psyche will have societal benefits.
But even if that's the argument, that's not the way OP phrased his point.
His point entirely about how the greats where just a product of their environment and not their individual greatness.
Not a discussion that seems to in anyway advance to a better progress of group or individual; but more about pettiness tbh.
I have actually seen this in many places.
One example on top my head- my neighbor was a talented video editor. Not legendary level, but okay. He eventually set up shop for editing wedding videos and such. He got contracts, he got praises and contracts in affluent families- all of this was not enough for sustaining a middle class life for an adult. He eventually set up a convenient store with the help of his father in law. He was quite old and had to move on to earn a middle class income. He did not have access to internet until early 2010s.
Now, his nephew, who is also my neighbor is also a video editor. But he got internet since he was born. He has all the resources. He has discord servers where people guide him and share resources. He is also an online gamer, and edited his own videos and posted them online. That led him to international contracts. Last month he earned US$120 working a single night.
That's HUGE for a 19 YO in small town India. Had his uncle (now in his early 40s) had all these, he would have made it big.
Circumstances do influence heavily how much a person succeeds.
I don't hold the line that equal opportunities lead to equal outcome- that doesn't happen.
But do think about the child that was born today in slavery in Libya, and a child born to a FAANG worker parent in Bay Area.
Do you think that they have equal chances of success?
And, I will also mention that Picasso had neighbors and peers. They didn't become as famed. Talent and individuality matters, too.
So its good to keep these concepts in the general awareness.