Google's John Mueller said something obvious to most the readers here, that "BERT does not affect indexing." He said this on Twitter:
Bert does not affect indexing. If you're seeing issues with indexing, that would be separate. I'd recommend checking in with the webmaster help forum, posting your site's details there.
— 🍌 John 🍌 (@JohnMu) December 3, 2019
BERT launched as a way for Google to understand queries better, which can impact how a site is ranked in Google search because Google might match the queries to the content slightly differently now. But it does not impact how Google indexes the pages on your web site.
So if you see pages of your web site dropping out of the Google index, it has nothing to do with BERT. It is always fun to blame BERT, I mean, he is the responsible Sesame Street character - but it would be inaccurate to do so.
Forum discussion at Twitter.
