When you are parking, you have much more visibility of oncoming traffic, and the other drivers see you more clearly too. When backing out, you have less visibility and are less visible.
It’s a terrible article. But backing in is absolutely the safer option.
You can't extrapolate from their situation to everyone's just because they "park cars all day."
Also, valets are more practiced and often are parking in areas where only they are placing and moving cars. So it's a more uniform, controlled environment. The cars are probably lined up better, making it easier to get in, and the other cars are probably not moving at the time since only the valets move them occasionally.
Personally find backing in much easier that the opposite. You have far more control and manoeuvrability at low speeds when your turning wheels are at the “back” of a vehicle rather than the front. It takes very little time to get used to using mirrors etc when reversing, and then becomes trivial to put cars into all manner of awkward spots that simply impossible to get into when driving forwards.
You can't practically reverse into a spot when someone drives too closely behind you and prevents you from reversing, or when someone noses into that spot that you appear to have just driven past.