If refugees could only claim asylum in the first safe country they reached, then countries neighbouring conflict zones would have to deal with all of them. Indeed, Turkey has 3 million refugees, Jordan has 3 million, Bangladesh has 1 million etc. These are not super rich countries in the first place.
One of the points of the international asylum system is to spread this load more equally globally, and that can't work if you are only allowed to seek asylum in the first safe country you reach. That's why it's not a requirement.