I'm worried that your question suggests a serious misunderstanding.
is it enough to prove that the entity is the owner of a public key by signing a certificate-request with their private key?
It is certainly not sufficient to have the entity sign something with a private key. How would the CA know which public key to use to verify the signature? It has to trust the entity to provide it. So anyone can contact a CA and say:
"I'm microsoft.com, and here, I've signed this with my private key. You can even check it with my public key. Now, could you certify that this public key belongs to microsoft.com, please? I'll pay you $1000!"
(In fact, this step is necessary, just not sufficient. If I didn't sign something with the private key, then I could send any public key - even Microsoft's - to the CA and ask them to certify it as mine. Then I can claim that something signed by Microsoft (maybe a patent application) was actually signed by me! So the CA will certainly check that I have the corresponding private key before it certifies the public key in the certificate.)
So the question is, what can a CA do to verify the identity of the person requesting the certificate? Nobody at the CA has ever heard of this entity before!
A simple option, that can be entirely automated at the CA end, is for the entity to provide an email address. The CA will send a challenge (such as a special URL on the CA web site with a long random number in it) to that email address. If anyone makes a request to the server with that URL, then it is presumably the person who owns or has access to that email address.
You can try this yourself, if you go to Verisign and request a free trial SSL certificate.
The limitation is that the only relationship that this certificate can attest is between a public key and an email address. That might be enough for some people, but it's not enough for everyone.
If I want to be confident that a particular certificate (or the public key in it) belongs to say, XYZ Inc of Elbonia, because I'm about to send some sensitive business details to them, I want more than just an email address. I want to be sure that the CA has done some serious probing. The CA should receive the request on letterhead. They should contact the company using a phone number from the telephone directory. (An imposter will have to fool the telephone people as well.) The CA should check with the business name registration office to check that XYZ Inc is registered at this address. They should post a document (with a long random number in it) to the registered address. (That means that an impostor would have to intercept the mail as well.) The CA may physically attend the registered office of the business of the organisation to confirm that the request has been made.
All of these identity checks are time consuming and expensive. A CA will charge the entity for such a service. But, if the entity wants to provide its clients with a high level of confidence that this public key really belongs to XYZ Inc, then that's what has to be done.
As Iain Collins suggested, CAs can provide cheap services for those who only need a small degree of identity verification, and expensive services for others. The certificate that the CA offers will contain an indication of the level of identity verification that has taken place. A person considering a transaction using the certificate can look at the CA's Certification Practice Statement to understand what this level means, and the types of identity checks that have been carried out.
Finally, the CA is not making any assertions about the entity's trustworthiness. Remember that a certificate is a link between a public key and an identity. The checks that a CA does simply ensure that the public key really is related to that particular entity, not someone impersonating that entity. The entity could be quite evil!
In summary, you can't use public key cryptography to determine someone's identity until after the certificate has been issued. The certificate says that the CA has used some other form of identity check, and allows other people to rely on that check.