What is the point of separate links to the download and its digital signature I see a statement like this on a product's download page:
"The signature for setup-x86.exe can be used to verify the validity of this binary using this public key."

where each this above is a hyperlink.

I thought the purpose of a sig is so that you could verify the validity of a document.

So what's the purpose of such a statement?

Surely it can't be  a protective measure just in case the site has been hacked to point the download link of the exe to a malicious file? In that case, won't the links to the signature/public-key themselves be altered by the attacker?
They don't expect that everybody uses their infrastructure. So, someone, a Man-in-the-Middle can change the executable or you are downloading it from alternative download sites that can contain malicious code.

One way to make sure that the file is not tempered is publishing the hashes on their website. Hash is very common for ISOs of the Linux distributions. This protects as long as you can access the hash values correctly. Another method is signing that also gives us information about the signer.

In the case of their website is hacked, you may be very unlucky to be in the middle of the attack that we expect it is a short time. If, however, they use a chained certificate and you know it before you can still check the signature. The attacker to be successful must somehow access the signature key. In general, to be on the safe side, the companies sign the files with an off-line computer so that the signature key cannot be accessed from an online attacker. As usual, the corresponding public key can be published.

So, in short, if the signature key does not belong to a trusted key chain that is verifiable, don't accept the file. Everybody can create a file with a signature and publish it with a verification key.