So other people have to do a bit of work (copypasting the license into the source) to exercise the permission you've granted to them (because that's what these licenses are). That doesn't invalidate your giving of said permission.
This isn't about subsequent redistribution (and if it were, I'd agree with your skepticism). The primary distribution must also follow the terms.
You can't cherrypick which parts of the license you wish to follow.
Why's that a problem? If you own the copyright, you have the right to distribute your code, and you have the right to grant other people permission to redistribute your code under the MIT license or any other license, whether or not you yourself are complying with the terms of said license.