...However, reopening the account is manual, and in that process American Express looked at your credit score.
I'd like to add that even if AMEX were to re-open your account, they'll charge you a fee to do so.
Do spend some time getting that credit score fixed. If indeed the only negatives are those three returned payments, your score ought to be way above 700
Once you hit 700 you should be able to get approved for any card as long as you don't have too many recent inquiries or new accounts opened. The credit score itself doesn't need to be fixed, just whatever the computer looks at when it looks at the OP's credit report... Could be there's a fraud alert on it or something.