They do that at a minimum. OV and EV certs require more work and do verify who you are (for some definition of "who") and that's where the more expensive CA's add value.
I understand what the browsers do today and I don't believe this help protect people. It should be very clear what type of transport security is in use, along with a score and what type of identity has been verified and what that means. I honestly don't know the answer to this, but I do know the existing methods in browsers just doesn't give a clear picture to non technical people.
Strangely, our CA called our human resource department to verify a lot of the information. I guess its all in who you chose.
Also some (maybe you don't think they're major?) don't offer DV. After all they can't be cheapest or fastest so why not focus on a product with higher value.
LetsEncrypt does this automatically, for free, and in a more user-friendly way. For information on the security considerations involved, see [1]. These are similar considerations to those of most DNS-based CA verification methods (which is most of them).
[1] https://ietf-wg-acme.github.io/acme/draft-ietf-acme-acme.htm...