Oh - I thought clients did. Was there some separate kind of cert pinning that clients checked that disappeared because nobody could get it right?
There's no reason clients couldn't, as far as I know (just like the various anti-spam measures where clients check for records that indicate what servers are allowed to send mail from a domain and consider it spam if those records don't match), but I also don't know of any clients that actually do. And a CAA record still doesn't actively prevent issuance of a certificate, which is really what would be needed.
There used to be a pinning scheme called HPKP, but the trouble with it is it's very easy to accidentally (or via malicious third-party action) end up in a situation where your web site becomes unreachable by people who've visited it previously because the key changed and the pin prevents their browser recognizing the new key as valid:
Clients cannot because the current CAA spec only requires that it be checked/valid at issuance time.
Thus, as a client, it is impossible to distinguish between:
1) A certificate that was issued in violation of a CAA record
2) A certificate that was issued correctly for a website that later changed their CAA record
There's also the basic problem that an attacker who can serve you a falsified A record for a domain can also serve you a falsified CAA record. This is why CAs are required to use "multiple perspectives" when validating domain ownership before issuing certs, whereas web clients generally don't have access to redundant infrastructure like that.