Coming from a company that had no dedicated QA team…I have a few thoughts on this matter.
As I understand it, with Shape Up the “QA” phase is done by the development party (the team that develops the feature/fix). Considering that the development is done within some kind of restraints of the existing environment, I’d say that this team should add whatever steps are necessary to the QA process as well (automated tests, “manual” tests, system tests, integration tests, etc). And those tests have to be performed, it is the responsibility that moves, maybe, from a dedicated QA team to the dev team.
Using Jira and Zephyr (hate Zephyr btw) is definitely not a necessity. It’s just a tool. I assume the departing PM and QA are saying it’s bad to leave Jira because that makes them feel their work was for nothing. I wish more companies stopped doing things because “they’ve invested resources” and started doing them because they should.
Following the same logic, you’d never refactor code because of the huge investment of writing the original code in the first place.