I was about to report this issue, but discovered the solution so I thought I’d post the solution in case others search for it, since searches show it has been reported multiple times in the past:
After I log in to nytimes.com with my paid subscriber login, with Brave Shields up, it is as if I have not logged in.
It turned out that for some reason I had the following non-default setting (in the advanced view when I clicked on the lion icon by the URL):
instead of the default which is
Cross-site cookies Blocked.
After I changed to the default above, all my login woes were strong, good looking, above average, woes begone.
I have no idea how this got changed since I never change that setting, but I’ll chalk it up to natural mellowing agents.