Keychain Won't Remember Router Administrative Password macOS High Sierra, Safari 11.1.2, MacBook Air mid-2013, Netgear R6400 wifi router. All the software on the computer and router is up to date. Both the Mac and Safari work without difficulty.

The network works fine, the computer connects easily (either via wifi or Ethernet), and except for this one issue I can access the router's administrative web pages and change its settings without difficulty. 

The issue is...Keychain won't remember the router's administrative password. The router's administrative login page has a box to check to "Remember this password," which I dutifully check. The password is remembered for a few minutes only, then I have to manually enter it again. It's a complicated password, and I'd be happy if Keychain remembered it, as it does for every other password I use.

I've quit and restarted Safari, deleted the Keychain entry for the router (and all other 192.xxx.x.x URLs, there weren't many), deleted Safari’s cache and cookies related to this URL, quit and restarted Keychain, restored the router .cfg file from backup, and restarted the computer in Safe Mode followed by deleting the Keychain entry again. 

The forgetfulness continues. Any idea why this is happening? Anything else I might try to fix it?
The 'forgetfulness' is likely to be intentional.

Safari won't offer to remember any passwords etc if the page tells it not to.
I don't know the technical details of how this is accomplished, but banking pages etc won't do it either.
The 'remember' box on the page is not for Safari to remember, it's for the remote device.

There used to be a freeware extension called 'autocomplete' that would disable this [lack of] functionality in Safari, but it hasn't worked so well in recent years.
I just discovered the author now has a paid version, called StopTheMadness, still fully supported, which might do the same thing. I haven't been able to fully test it yet - just got it 5 minutes ago.