No, actually not. He has the student say the lines or the spaces. Not the same thing. And he uses letter names, not solfege (although he does say solfege should work).
You've proved my point - you didn't understand why his method works.
(First, I already acknowledged he uses letters. I'm merely showing a parallel between that and how some of fixed-solfege, as a note-reading method, is trained.)
Read the first two steps in his post.
Even
before we get to any training with the staff, mental fluency between the letters is achieved by being able to loop stacks of thirds. Memorized, like a phone number, and starting on any letter. He has the student memorize EGBDFACE, and then be able to do that starting on any letter.
The entire method is based on being able to stack thirds, and then later connecting that with the staff by recognizing line-to-line and space-to-space are thirds.
The part you might be confusing is that he teaches a beginning student to recognize the grand staff as a whole, but that's a completely separate issue to what we are talking about.
The part I'm abstracting from his post is the mental fluency involving the diatonic nomenclature with intervals. It's the ability to quickly spell intervals and mentally manipulate them. He only does thirds, because that is sufficient at that level. However you can expand that later on and connect that with the visual presentation on the staff.
Mentally being able to switch clefs requires you do understand relative and intervallic positions on the staff and NOT absolute. It doesn't matter whether you use letters or solfege for this.
I only mentioned fixed-solfege because it's ability to be more easily verbalized like a "memorized phone number" is what makes it easier to use then letters when you do more intensive training.