How to buy a ticket from France to the United States if the person flying won't have any proof of payment?

If you are worried about showing payment, you can eliminate this by going to a travel agent and have them book the flight. That automatically verifies FOP for the ticket. You can also visit the airline's office and do the same if they serve your home town.

But be sure to talk to your friend by phone BEFORE doing anything. They can call you collect. The lost cards, cash on vacation email scam has been going on for years.

he might not be able to board the plane if he doesn't have the original form of payment.

I read the linked answer and it does not mesh with my experience after hundreds of flights I took in Europe, the USA, Canada and in between. Consider people flying on the dime of some company. The passenger certainly won't have the card for that.

Even in the case of IRROPS (irregular operations) noone will ask for it. I've had an Air France Budapest-Paris flight which got cancelled, AF put me on a trip connecting via Munich (btw this is the example I use for flying traditional vs low cost airlines) and never asked for proof of payment -- it's not like I had one...

My experience, without exception is: I show up at check-in, present my pile of passports and relevant immigration paperwork, get a boarding card, go through security, board. If I am asked for a credit card it is only because I need to pay for checked in luggage.

So I would just offer buying a ticket and be done with the problem. If the friend insists on receiving money then it's a scam.

Airlines in Europe usually ask for your payment card at the airport if and only if the cardholder is travelling. I routinely book flights for my parents (directly via the airlines' webpages, never via reseller) and so far there has always been an option to indicate whether the cardholder was travelling or not. Note that some airlines (Lufthansa if I remember correctly) require you to log on with your frequent flyer credentials before you can book a flight for another person or to go via their phone booking service.
If in doubt I would always call the airline, they are generally able to sort out non-standard bookings fairly efficiently (once you are past the waiting muzak that is...)

As simple as it sounds, book a ticket which doesn't require FOP verification. For example, here's a reseller which doesn't require you to present the credit card for verification at the airport (and even accepts virtual cards).

You may even book the ticket with an actual virtual credit card to be sure.

Some airlines (Qatar, for example) provide a form for payment verification, if the credit card holder is not traveling with the passenger. One needs to upload the monthly invoice from the credit card company to prove that he owns the card/has access to the invoices which are generally accessible only by the owner.

Another alternative that I found online when I was in the similar circumstances, was to sent a copy of your invoice to the passenger via email. They can take a print of it if asked by the airline authorities. This proves they have access to the invoices, which, also implies the point mentioned above