The low quality of the voice assistant meant they doomed the Cortana name. It got associated with garbage voice assistance, so they had to rebrand.

If there had been consumer good will for voice assistant performance before poisoning the name, it might have survived... but when you first product in that realm is garbage and it lingers for a long time, you're dooming the name.

Also, synergy with Halo is meaningless to Microsoft. Most folks do not care or even know about Halo.