Fix Interactive container hangs when redirecting stdout

Fixes #6509
Closes #7658

This is a rebase of #7658 + tests. Thanks @VojtechVitek !

I even tested the `events` test with one commit before the #5979 PR and it failed. So all 3 are needed.

Ping @erikh @jfrazelle
The point was not to rewrite the current API. The client pkg might be included in the 3rd party code, right?