A hardening fixer cannot hurt Kodak, Fuji or Ilford films, but it is rather useless as they are already quite hard. OTOH, a hardening fixer can work well with some other films on the market. The new hardeners were patented long enough ago that the patents have expired, but the technology to use them in coating is very complex and therefore has not been implemented by other than the big 3 (AFAIK).
Kodak makes their hardening fix with the hardener in a separate bottle. Its use is optional.
Use of a hardening fix helps prints more than film under ordinary conditions, but at one time when film was just thrown into a tub or sink and washed, there were scratches. So, hardening fixers became rather normal for all products, but are now optional for products from the big 3.
Alkaline hardening fixes have been sold by Kodak going back to the 50s and 60s. You can build alkaline hardening fixers, but you cannot use alum.
BTW, treatment with some chemicals after use, will reverse Alum hardening, but no known method outside of the lab can reverse the new incorporated hardeners. The reversing method can also dissolve skin which is a protein as well, and this method is rather dangerous.
A hardener in the process at least helps reduce swell if that is what you want, but at the same time it slows fixing and washing.
Kodak (AFAIK) recommends no specific fixer pH and as noted above, hardener is optional. Color images however, are best kept either neutral or at pH 4..5, no lower.
More than you ever needed to know!
PE