Brahms is dead and has been so for some time. But, yes, the standard stuff is included later or reserved for later study with teachers using various methods and materials. Rhythm first, then pitches. We cannot learn three or more things at one time. We do best what we learn first.
Fluency is dependent on a sense of keeping up with a forward motion.
A forward motion in time is based on beats and meter, not on pitch letter names. So, ultimately, the development of fluency has most to do with flowing with the speed of the rhythmic musical information and dealing with it while in motion. A good canoe paddle technique and arm shape are rather pointless if your canoe is now stuck atop a log in the middle of a river because you never learned to look downstream and steer smoothly. We hear the scale degrees and we respond with trained finger patterns. Additionally, these do not need to be understood in order to read, play and hear tunes at different pitch levels.