Trying to buy Apple Care, I get response my product does not qualify for it.

If you purchased within the past 60 days, you're eligible. However, if you bought from a 3rd party through Amazon's marketplace, it is possible that the seller is not an Apple Authorized Reseller but rather purchased the Mac and re-sold it to you, in which case the recorded purchase date is when the seller bought it, not when they sold it to you.

Yes, or it's more than 60 days from date of purchase (too late to buy AppleCare.) Amazon offered you an Amazon warranty for around the same cost as AppleCare. You may still be able to get that from Amazon.