I’ve done both: in scenarios where we book, I ask the customer if they have preferred flights (and FF numbers) and work with a travel agency to set it up. The ticket is issued to the customer, so they can make modifications, check in on the app, et al.
And Rebecca— yikes, I’ve been there. I know this is a big pain and not the best experience for your customer but hopefully it was smoothed over and now you have oodles of ways to avoid it in the future ☺️