Is mailing yourself a copy of your work a legitimate alternative to registering a copyright?

This is called the "Poor Man's Copyright" and its utility is dubious.
First and foremost, there is no substitute for registration. In its FAQ, the U.S. Copyright Office says this about it:
There is no provision in the copyright law regarding any such type of protection, and it is not a substitute for registration. https://www.copyright.gov/help/faq/faq-general.html
Without registration, you cannot bring a suit for infringement. Without timely registration, you lose access to statutory damages and must prove all alleged damages. Relying on a Poor Man's Copyright could cost you dearly down the road.
"Fine," you concede, "Registration is superior. But is mailing a copy to myself useful at all?"
Conceivably, but almost certainly not.
The lay-person's theory behind the Poor Man's Copyright is that by mailing a copy of the work to themselves, they will have a sealed, postmarked (dated) envelope, which may later be used as evidence of when they created the copyrighted work. The idea is that if there is an infringement action, they can whip out that envelope and show the postmark as evidence that they created the work first and therefore should win.
The problem is, this notion misunderstands how a copyright infringement action works.
Without getting deep in the weeds, evidence of priority alone is not enough to find infringement. You can pull out your postmarked envelope, rip it open, show the court the copy of your work, and say, "Look! I sent this to myself months before the defendant created their work," but you've offered nothing conclusive.
For one, there is a concept in copyright law called the Independent Creation Doctrine. This doctrine stands for the idea that two identical works can be created independently without infringement. I may have painted a picture of a blue circle on a white background, for instance, and, three months later, you may have painted an identical picture of a blue circle on a white background, but that does not mean you infringed on my work. What if you had never seen my painting? You can't copy something that you aren't aware exists. And this is why one of the fundamental elements of an infringement claim is…well…proof of actual copying.
If I allege that you copied my blue circle painting, and you say you didn't, the only way for me to prove that you did is by offering circumstantial evidence (which copyright law happily permits since we all know people are liars). One essential piece of such evidence is proof that you had access to my work or, as some courts have said, an opportunity to copy it. If I can't show that: no infringement. I lose. And my postmarked envelope? At best, it was useless. At worst, if the only copy of my work was sealed up inside that envelope this whole time…might be hard to argue that you had access or an opportunity to copy.
Furthermore, a court may be suspicious of the reliability of the Poor Man's Copyright. In two separate cases out of New York, the courts determined that an envelope used for that purpose was tampered with after it was postmarked.
One of the only conceivable uses for a Poor Man's Copyright arises if you find yourself being accused of infringement. If I accused you of infringing upon my blue circle painting, and you could show an envelop containing a copy of your blue circle painting postmarked before the date I purportedly created my painting, it could have some probative value as part of my Independent Creation defense. But that's about it. And that's a longshot.
The moral of this piece: don't rely on the postal service to protect your intellectual property.