I came to post the same thing. You have a stick from 12,000 years ago, and a story from 100 years ago, and they link them because of a similarity "there is fat at the end of the stick".
You could get fat on a stick by roasting a small mouse (so a small hearth), or tons of other ways. This really isn't enough evidence, not without a ton more findings of the same thing at various dates over the timeperiod.
If tiny hearths with fat-covered sticks were common elsewhere, this would be a valid objection. But as it is, this seems to be a distinctive practice that is present at the same ___location separated by 12k years, and the ancient one was buried and so not observed for most of that time. What's the alternative explanation except a common root in a cultural practice?
You could get fat on a stick by roasting a small mouse (so a small hearth), or tons of other ways. This really isn't enough evidence, not without a ton more findings of the same thing at various dates over the timeperiod.