The answer to this question is that Shir Hamaalot/Al Naharot Bavel are not "prefacing passages" to Birkat Hamazon, but are actually supposed to be recited during the meal itself, prior to beginning Birkat Hamazon.
As noted by the Shelah (Sha'ar Ha'otiyot, Kedushat Ha'achilah 170; some cite it as page 82b, such as here), the root of this practice is ...
Halachipedia has a good rundown and explains as follows:
When to make the Bracha
According to some it is preferable to say Shehecheyanu before the Bracha on the fruit, however, the minhag is to make the Bracha on the food and then the Shehecheyanu.1
One only makes a bracha on the first time one eats the fruit (that year).2
If one forgot to make the Bracha ...
As an update about this question:
Recently Rav Dov Lior published a shiur (available here: https://www.youtube.com/watch?v=veFtVA9dnh4) in which he says that fruits/vegetables which producers produce with intent of the production going at least partially for the making of juices (אדעתא דהכי), juice made from them, if there is no other things mixed on it, ...
Rav Chaim Kanievsky shlita recently celebrated his 93 birthday K"H. He reportedly told people not to wish him a happy birthday becuase " Pharaoh is the only one in Tanach who celebrated a birthday"
I also heard this in person from Rav Neventzal shlita.