By the power of truth, I, while living, have conquered the universe.
I pretty much agree, Cortana dying does essentially 2 things:
1) Beautifully reinforces the thematic direction Halo 4 took with the whole conflict of accepting mortality, and the separation of machine and man - the ultimate discovery by John that his humanity, as brought out by Cortana, is his greatest strength. The "tools" Halsey supplied to "maintain that efficiency" portraying Halsey as the 'creator' or 'preserver', much like the Librarian.
2) Sets up Halo 5 and 6 to be meh on a character front for the Chief. Despite how effective Halo 4's themes were, Cortana is ESSENTIAL in John's story. Halo 5 and 6 just wouldn't be the same without her, much like how you pointed out that Halo 3 was pretty poor on this particular front.
Killing off Cortana has 343i backed into a corner in a crumbling building, if they let her live they backtrack an essential thematic element of the Halo series, but if they keep her dead I feel that the characterisation of John would suffer. I see Cortsna as an extension of John himself, if she dies, a part of him is truly gone - that hole can be filled only by melancholy writing.