I don't have any suggestions, as I am not super-familiar with the current situation, but I can understand (and suffered) from some hidden bugs (not immediately obvious) taking some time to surface and/or reach the right people, but I have some questions:

* How often are issues surfaced in the group0 -> group1 vs group1 -> group2, are there any stats to back the need for a change there?

* Without changing the actual deploying days or the frequency, would there be any benefit of shifting the deploy over multiple weeks? (random example Tu: group1->group2, (new branch) We: group0, Th: group0-> group1) or would that make things worse?

* You mention commons. I am guessing Commons, and Wikidata, to some extent- are both large sites with a lot of visibility but also very different from the core features that are similar to most other wikis, but the test version of those on group0 may not be enough to catch all issues. Is there something that could be improved specifically for those sites?

* Can we do something to improve the speed from "a user notices an issue with the site" to "the right team/owner is aware of it and acts on it"?