This is a very very big subject that touches every area and aspect of the oil industry. I will answer it the best I can ,off the top of my head , so there may be an area or two that isn't perfect.
Amsoil unlike other oil companies is family owned and sets out to make the very best oil it is possible to make, this is how a small (relatively) business can survive and thrive. It will buy the very best base products from whoever has them and will change suppliers if there is better on offer elsewhere. This is a very critical starting point as it allows Amsoil the flexibility the Multi Nationals just cant have, they cannot , darent, go to their competitors for 'x' because what they have is not as good; would look good in the press I'm sure!
This leads to the API testing, which the Multi Nationals back for what will become clear in a minute. The individual tests cost about 1/3 million I am led to believe. So the bigger you are the easier this is to handle. The Muti Nationals generally sit on their laurals and generally using their own products, and the cheapest combinations necessary to meet a given standard, make the very same oil for long period irrespective of improvements that could be made. They are actually made to a price Amsoil makes continuous improvements, it has this thing about producing the very best, being top on each of the 10 recognised international tests. Every change would necessitate the retesting of the product for API purposes. (another 1/3 million not to mention the appx 6 months this would take) And of course some times updates happen faster than the API could test anyway. I could cynically suggest that if a supplier made a product 'unavaliable' to Amsoil then they would be forced to reapply for API testing with the 6 month lead in and cost, perhaps you can recognise that this could lead to a smaller business being totally controlled and to loose its freedom to go its own way.
This should now be painting a picture.
The ZDDP bit is very similar. Whilst it would be correct to say that ,all things being equal, the more ZDDP there is in your oil the more likely you are to have premature cat failure. But all things are not equal. The testing evaluates oil by amount of ZDDP with a safe maximim for cats survival using an 'average' oil as a base line. (I think this is correct rather than worse case senario) Amsoil turn around and say that there oil is not 'average' and that their tests show that their oils put out less ZDDP than others that have passed. They still however have too much ZDDP to pass the Muli Nationals backed API test. Amsoil follow by saying that its the oil being burned off that causes the ZDDP to be in the exhaust not the quantity in the oil. Amsoil has the lowest volitility of all oils so it follows that less is burned off and the emissions are therefore low, test our oils . The API just says Amsoil has too much ZDDP in it Full Stop.
So your choice really, do you want the very best oil that can be made, that has lots of the expensive ZDDP in it to prevent wear and protect your investment and which although expensive initially lassts longer and can actually pay for itself and save you money, wont harm your cats but doesn't have API certification because Amsoil have been at war with the API for years. or do you want the alternative. Your car your money your choice