could not a 'straw purchase' only be when you first buy the gun through an FFL
I believe so. Leaving aside the issue of whether the recipient is allowed to receive a firearm or not (since it's already been established that that's s separate issue) I don't think there is any law against acquiring a firearm on someone else's behalf. At least, I couldn't find anything like that in the Gun Control Act.
Of course, I'm not a lawyer or a cop so please don't take this as advice