FYI, @corona10, this is the largest part of bpo-40956, and I see that it might be too much for one review. Let me know if I should break this PR up in multiple parts.

@vstinner cc @erlend-aasland
I am quite busy until next week, Can you please take a look instead of me? ;)

Let me know if I should split this up; excluding the clinic code, the short stat is 3 files changed, 292 insertions(+), 210 deletions(-). I'll be happy to split it up, if that can help.

All right, how about this: I'll split this up in two parts:

1. All the straightforward methods
2. Connection.execute*() and Connection.backup(), which require small modifications in addition to AC

The former, with the larger line count, should be the easiest to review. The latter may end up with adjustments; I may not have chosen the best way to resolve the issues encountered.