All tropes have a genesis, in the fantasy realm for the Half-Elf Ranger that genesis from, what I can gather, Aragorn from Lord of the Rings. From here the H-ER (Half-Elf Ranger) seems to have become a Fantasy trope that is placed into the genre with reckless abandon. I do admit that adding a H-ER makes sense from the point of view that Half Elves do not fit in well with either Humans or Elves. With the humans they live too long to fit in to society and with elves…well that is a whole different post but most Elves are fairly snobbish when it comes to their bloodlines and do not like half-elves. Even if that were not the case most Elves live to be 500+ years.
This plays out through many different Fantasy lines, such as Tanis Half-Elven, who first appeared in the book Dragons of Autumn Twilight that was the first book of the Dragonlance setting for D&D. Granted, in these series of books we are essentially dropped into the middle of a story that unfolds as the “War of the Lance” we learn that he is born of an Elven mother by a human father, the accounts differ on the circumstances of his birth, however either way he was an outcast to the elves for most of his life and as a Ranger, in principle if not in canon.
It is not to say these types of characters do not have their places, however the use of them as part of the core “hero” or “saviors” groups tend to be played out. There are many other ways you can incorporate “outcasts” or even Half-Elves into a story setting without falling into a trope of the H-ER. You could use a character who is a ranger because they had an issue in the past caused them to leave society, this can be good as you could discuss it with the GM to use that to draw the character into situations where they would be forced to either confront their past with a revelation to the whole group or work against the group to try and keep it a secret.