A smaller range certainly has an influence to the writing because one tends to write what one can sing. However, it is not clear to me how much of a smaller range Jon has. When the voice gets lower it doesn't need to influence the range.
Topics chosen (God Bless This Mess) and production styles chosen (to cover weaknesses in the singing) are more connected with how the person experiences his/her vocal decline than the effective limitations. But there are many other reasons for making certain choices. So I'd say there is an influence but a minor one.

Maybe the influence on the listener's experience is bigger. Forced singing just doesn't sound great. The same melody in a lower key may have less impact, because higher tones stand out more. A lack of power in the voice also doesn't work so well with certain types of songs (Maybe Jon still writes to less for his nowadays voice, so maybe the influence of the vocal limitations into the songwriting should even be bigger?).

Just some thoughts of mine.