Jira tracks almost every change automatically.
If you have stuff that is not writing to the history, then you've got an app or customisation that is deliberately breaking it. Remove or disable that, and you'll find it writes everything (exceptions that spring to mind are issue links and re-ranking, but every single field change is logged)
You could restart Jira in "safe" mode, with all of them disabled, then turn them back on, one by one, until the problem happens again. Otherwise, yes, you'll have to turn them off manually.
I would also look at the system information page to see if there are any files that have been changed in your main installation. (It should always report "jira-application.properties" as having changed, that's normal, but are any other files mentioned in that section?)
Also, can you tell us exactly what is not being recorded? And what the history of the issue looks like?
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