The phrase “to be in the groove” has no connection with being in a rut, for the current crop of American young people use it to mean to be exactly right, to fit exactly the mood or spirit.
It seems to be a coinage of the jazz or swing era of music and to have been derived from the phonographic records of that music; that is, to the quality of accurate reproduction of such music through a good needle traversing the grooves of a record.
The phrase is not more than about ten or fifteen years old, and, although applied generally to things that are functioning smoothly, its specific application is music.