A great additional point. That alone would probably be a much greater factor than any aging-related detriments to Halsey's brain when it comes to making a new AI to challenge Cortana.
To add on, even if Halsey underwent the same process to make another AI like Cortana, I still fail to see how that AI would be able to go toe-to-toe with Cortana considering she has the power of the Domain at her disposal.
Conditions wouldn't be the same, though. Halsey is older now than she was when she created Cortana. Brains get worse with age, as do most organs. A flash-cloned brain from current Halsey probably wouldn't produce as strong a matrix as her brain from ~10 years ago. Also aging as a risk factor for disease starts increasing exponentially after middle age (~50), so there could indeed be a big difference between Halsey's 57 year old brain and her 67 year old brain.
Catherine Halsey sacrifices herself to save John and Blue Team by turning herself into an AI with a new name that looks like Cortana, only to beat Cortana and save the galaxy. My theory.Maybe she doesn't have to kill herself in the process; Cortana is the only A.I. created from living tissue, so Halsey could create another Cortana under the same conditions to combat her evil doppleganger.
The only conceivable path I could see in them getting a fresher mind is if any of the extra cloned brains Halsey put in storage on Reach were recovered. But that isn't very likely since:
A. The Covenant glassed Reach
B. The brains would likely have been stored at CASTLE Base, which was destroyed by Dr. Halsey
C. Assuming the area the brains were in somehow survived, it'd be cut off from power to the facility and any cryogenic storage used would fail
D. The brains would have been sitting around for 5-7 years by the time Infinite begins, a problem if C is correct and nothing remains to prevent the brains from decaying
I mean, we should stop at B, but even then there is very little to suggest collecting the brains would even be possible.