In general, no. Tick the box to indicate the payer isn't travelling so they know the card won't be available at check in etc. Even if that's an almost wholly theoretical consideration.

There was a recent thread involving QR that had some difficulties in this area related to overbooking, but since you have access to the collective FT knowledge, you could deal with that unlikely situation without much difficulty

I assume this is to get the Amex Avios, I'd say book away. It's hardly an unusual occurrence, many businesses will have travellers booked on the bosses card etc.