The standard for credit card sales now is that you get the money 2 business days after the transaction. Viewing my client’s sales records and credit card deposits, this is basically always the case. Sales on Monday get deposited Wednesday. Sales on Friday through Sunday get deposited Tuesday, or Wednesday if Monday’s a holiday.
Regardless of whether it’s possible for people to get their sales deposited the same day, this is how the industry works now.
The economics of payment in the 1950-1980s were very different, so it’s quite possible that Amex made money just being a charge card then, but no longer would now.
I expect that if they still have a charge card offering, it is mostly a loss-leader that’s establishing a relationship with people who are not interested i credit cards.
It’s not risk free. I’ve never paid credit card interest or a late fee, but I have had my card used fraudulently several times over the years. Assuming that the merchant followed the proper procedures (which are very limited to reduce payment friction), the credit card company eats those losses. There are other risks I’m sure, that’s just the first one that comes to mind.
By the 80’s, it was common that the slips were collected and flown down to islands in the Caribbean, where they were processed. The results were transmitted back. One common job back then was data entry, a rote job that was easy to train people for - but fast efficient keypunchers were in high demand. It was convenient to have a place to do data entry where wages were significantly less that the USA.
In that setup, even Amex should have had no problem with a rapid turn-around. I don’t remember it taking more than a week, tops, between when a charge was made and when it showed up on the bill. (If your CC bill cut-off was say, the 10th of the month, you would see charges you’d made 2 days before. Even allowing an extra week before they printed and mailed the bill, that was at most 1 week turn-around.)