All the primer books recommended (Faber, Harris,Alfred, Bastien,and Hal Leonard - I forgot to mention) begin with black key clusters moving all over the keyboard. The purpose of this is to give the student a grasp of keyboard geography and get them familiar with the keyboard. All of these primers do this.
Don't be discouraged about not being introduced to the staff yet. It is simply too much information for beginners to have to learn proper hand position, finger numbers, rhythm, and notes on the staff at the first lesson. That could be a sure way to set them up for overload and feeling as if piano is too difficult. Remember your student is only 6.
The staff is introduced after the student is comfortable with the other skills. I pulled out all the mentioned primers, and you're right, Faber does begin with boxing the finger clusters, instead of using notes ( the only one that does this). Their thinking may have been to give the student more time to process finger numbers and keyboard geography before introducing rhythm. 2 skills instead of 3 at one time. Are you using the lesson and Performance book together? I find it's helpful to have more than one book to study. The student receives more reinforcement that way.
Also, do you teach technic? Meaning 5 finger patterns and arpeggios (hand over hand)? These skills really help students get a sense of key as well as reinforce proper hand position and rhythm without having to read music. This will also help teach transposition. I have my students transpose their 5 finger pieces from the method books into familiar keys (5 finger positions that are similar to begin with, eventually being able to play them in all 12 keys).
If you're not happy with Faber, spend some time in a music store perusing the other methods and find one that works for you. I still stand behind all methods mentioned as they give the student time to process information a little at a time. Beware of a method that immediately jumps to fixed notes on the staff, within the first few pages (Thompson and Schaum). With the exception of Frederick Harris who teaches notes on a moveable staff, not fixed. Focusing on intervallic reading.
I still stand behind my choices and know that they work. I've used them all. Do spend time researching what works for you.
Good luck, and keep me posted on what you decide.