Refactor and clean up the automated test suite

Some stuff was done in #6543 (closed), but there’s a lot left to do to prevent it from becoming a crazy mess.

Subtasks

• #9030
• #9040
• #9223
• #5847
• #10283
• #10328
• #10329
• #10341
• #10429
• #10864
• #10974
• #11376

Related issues

• Blocked by #6094 (closed)

Original created by @anonym on 10237 (Redmine)