I wouldn't make the AMEX platinum your primary card for spend. If you're looking to amass points (membership rewards) verify that you're earning points on your green card. I have a corporate one and have the option to pay some fee to turn that on - your situation may differ from that. I think the green card gets 3 Membership Rewards points/dollar on travel/dining/transit. If you travel international frequently, see below.
The Amex Platinum is great for travel benefits (centurion lounge for one now, priority pass, hotel credit, airline fee credit, Marriott Gold Status, Hilton Gold Status, premium status with car rental companies, TSA Fee credit, Clear fee credit, etc). But this is not a great card for most of your credit card spend if you are trying to earn points.
The AA Executive World Elite card is good for two things - Admirals Club Access and loyalty point bonuses towards status. If you are flying AA regularly, get this card and then determine your temperance for multiple credit cards. You can make it real easy and use just this card for everything to focus on getting status. You'll earn AA miles but its a sub-optimal strategy if you really want to get into the miles/points world. If you'd prefer some more flexibility, get this card for its benefits and use it where it make sense. Then get another card for your regular spend.
Since you have the freedom card, I'd recommend getting the Chase Sapphire Preferred card ($95/annual fee). The cash back you're earning on the freedom card can be converted to ultimate rewards points which can be transferred to many travel partners for bookings (Hyatt, United, Southwest, British Airways, IHG, Marriott, etc).
So pick one of the premium cards (AMEX Platinum or AA Executive) based on your travel needs, but that's not your main card. Either point-a-rize the Green card if possible or get the Chase sapphire preferred. Or ignore all of this and just get the premium card and put all of your spend on it. But there's opportunity to maximize your earnings if you go the other route.