After all this time, I finally got DH to call AmEx to find out if he was sitting on a half million points (he wasn't.) He's had an employer-issued corporate green card for 21 years. He pays the bill, and the employer reimburses. Unlike many other members here, he didn't seem to have rewards.
The deal is that if he wants to pay $75/year, they will enroll the card in Membership Rewards. Alas, no credit for all the charges he's had over the years. He probably puts about $10K of charges on it a year, and the fee would be the equivalent of 7500 points, so he isn't going to bother.
But if anyone is in a similar situation, you can collect rewards if you're willing to pay.
My guess is that the employers have the option of choosing to let employees get points or not --probably a lower fee for them if they don't, or something.