Ideally, a French horn player plays just like he or she would sing, however one must sing properly. The use of syllables is quite important and generally we play with and open mouth cavity and breath deeply for the low register. And for the high register, we use a smaller mouth cavity and breath higher in the lungs.
By the way, A relaxed throat and generally body are quite important you definitely do not sing with a tight throat.
My main point here however is the use of syllables. I have never been told by a teacher to use the syllable “TOE”, but I like to syllable for the low register. And then generally, I try to keep my mouth cavity as open as possible for a big tone. “Tee” syllables in the upper register are understandable, but try to remain relaxed and always keep the air flowing.
I was quite surprised when I realized I could hold my breath or play a longer tone if I kept my nose relaxed. I have found that resonance in the high register is quite connected to residents in the nasal passages.
I will have more thoughts on tone eventually.