I no longer have paper receipts, but I have credit card statements that show the GC purchases on my Chase Freedom cards, so that's proof, right?
Maybe, maybe not. Your credit card statements just show that you bought
something there, there's no definitive connection to
those specific gift cards. It's probably unlikely anything will go sideways (indeed, you might be named Customer Of The Month or something), but if you trigger some adverse action, without receipts, you're in a pretty weak position to negotiate. Now, this is Amazon we're talking about here, a company that purports to treat customers like gold (as opposed to somebody like GreenDot or PayPal, who tend to treat customers like Donald Trump treats his employees).
Still...with $10K on the line, you must like to live dangerously. Way, way too risky for my tastes. If it were me, I'd be freaking out (and I'd have a shoe-box full of receipts).