While the process may appear straightforward for some, including those with a legitimate purpose, there will be a good number of more questionable situations, where some passengers will be unaware of the new rules, other will plead ignorance, assuming they are exempt, play the HIPAA card (claiming such information is confidential), creating a scene.
As 3Cforme states above, the implementation at DL and UA appear to have gone smoothly, so the risk of a major incident is low, good to see the industry coming together, creating common rules in this regard. I expect AA and WN to quickly follow, or risk becoming the emotional support animal carriers.