I am sure this is something that has faced many teachers, especially those of us that do clinics, or work with beginners. Why is there no simple fingering chart? Now I am sure there is one out there, in theory, but I really haven’t encountered one.
When I was developing my student horn book a major point of focus for me was an approach that was straight forward, since that is how fundamentals should be, simple and effective. Something I encounter more times than I want to admit is showing up to work with students and being faced with frustration over being able to pitch notes from written middle C to the first G in the treble clef. When they start playing I am sadly never surprised to find Fs being played on the open F horn, Gs on F horn 1st valve, and so on. You know, using B horn fingerings without using the thumb.
When I notice this, the first thing I ask is about a fingering chart, either they don’t have one, or they have one of several method books, I am always amazed with all of these method books that the fingering charts are never great. They never make an effort to differentiate between what fingering is for what horn, or it’s inconsistent. It must be rocket science. So I submit to the horn world the fingering chart from my Student horn book. A simple, no options approach to fingering. It is based around standard conventions, and doesn't give several options for each note. Though in time it is important that the horn player learns all the fingerings, but let’s keep the horse in front of the carriage.