Yes, Stu, you are missing something. Just as there are differences in how easy or difficult it is to finger, say a stringed instrument, in one key or another because one has to constantly jump around from position to position & string to string, or, say, a trombone can't do a glissando upward beginning on a note which is played with the slide really close to the mouthpiece instead of way out there, so also, voices differ.
One's Tessitura is not just about the range of notes which are better for one's voice, but also about how sometimes one's voice literally works better & sounds better in particular keys. While my voice can be lousy in the key of D, the same exact song will sound magnificent in Db--or at least it did when I was younger.