Make our test suite survive changes in the surrounding environment

Originally created by @intrigeri on #15953 (Redmine)

This is a placeholder ticket to track what’s part of the 2019-2020 roadmap on this front.

We don’t know yet what we’ll have to do among:

We’ll know better around January after we make more progress on #15460 (closed).

Edited May 15, 2020 by intrigeri

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