While I agree the commit message should summarize what happened in the commit, I pretty strongly disagree that the linkage between git and the ticketing system should not be considered an absolute "first class relationship". There is simply too much information/history to consider that linkage just a "nice to have".
Regarding migrating to new issue tracking systems, "not maintaining history" is absolutely a critical fail at that org. Every decent tracking system I can think of has ways to import issues from other systems - skipping that step is pretty inexcusable for all but the earliest stage projects. And the vast majority of them have ways to transparently map ticket IDs. E.g. in Jira if you migrate a project and give it a new project ID, the old IDs can still be used and they will transparently redirect to the new ID.
Regarding migrating to new issue tracking systems, "not maintaining history" is absolutely a critical fail at that org. Every decent tracking system I can think of has ways to import issues from other systems - skipping that step is pretty inexcusable for all but the earliest stage projects. And the vast majority of them have ways to transparently map ticket IDs. E.g. in Jira if you migrate a project and give it a new project ID, the old IDs can still be used and they will transparently redirect to the new ID.