So am I doing it all wrong? I use Tunes for Ten fingers, unashamedly in Middle C position until in book 2 when the C below middle C is introduced. Then we do pieces in that position from various books, then extending the hand position, then changing fingers on notes. It all seems a logical way of teaching children to read music. I find a lot of children get confused by all the different hand positions in PA book 1. And I haven't noticed hand tension in my pupils, in fact they're really relaxed because they love what they're playing.I grew up on Schaum which I imagine was considered a new, innovative method then. And I remember my mum realizing I was playing from finger numbers! But I was enjoying playing and soon realised that any note could be played by any finger.Maybe some of my pupils at first associate notes with certain fingers . But they move on from that, and at least they do read the fingering which is my bugbear with inherited students.
I personally haven't had any seen any problems with starting young children with middle C position but I am interested in learning more about the merits of different approaches - eg. Irina Gorin's avoidance of the short fingers that Helen mentioned above.
I have the same experience as you, Latin Pianist, that even if children do associate certain fingers with certain notes at the start it doesn't mean that they will in the long term.
Some really do have problems telling the difference between note names and finger numbers but I have a feeling that those who have longer lasting issues with it would also have problems with any other method.