Why macOS asks, and what to do when the switch is on but dtangle says it is not.
macOS lets dtangle see which app is in front without any permission at all. Reading the window title is the part that needs Accessibility, and the title is most of what makes a timeline readable: the difference between "Cursor, 90 minutes" and "Cursor, 90 minutes, on the file you were actually stuck in".
Without it dtangle keeps running and keeps syncing. It just records considerably less, which is the awkward part: nothing looks broken. That is why the app tells you rather than waiting for you to notice.
This happens, it is not you, and toggling the switch will not fix it. macOS stores a fingerprint of the app alongside the permission. If that fingerprint stops matching, usually after an update or a reinstall, the entry keeps showing your app's name with the switch on while the permission no longer applies to the app that is running.
Flipping the switch off and on does not rewrite the fingerprint, which is why it appears to do nothing. The entry has to be removed and added again.
The fix
The menu bar warning clears by itself. You do not need to dismiss it.
dtangle's Grant Accessibility menu item does the removal step for you before it opens System Settings, so from the app you only have to do the adding half.
Clearing the stored entry from Terminal is the same operation with none of the clicking. It only resets dtangle's own permission and touches nothing else:
tccutil reset Accessibility com.dtangle.desktop
Then reopen dtangle, add it under Accessibility with +, and switch it on.
The warning triangle in the menu bar disappears, and the warning on the Devices page for that machine goes away within a minute. If you want to be sure, open your timeline afterwards: entries should start carrying the document or page you were on, not just the app name.
Reads the title of the frontmost window, and nothing else. Not keystrokes, not screen contents, not other apps' data. Titles are truncated before they are stored, and you can delete a day's activity at any time. The FAQ covers the full picture.
Reading Safari's current URL uses a second, separate permission called Automation, which macOS asks about the first time it happens. Granting Accessibility does not cover it, and declining it only means Safari time is recorded without the site. Chrome and Firefox do not need it, because there the browser extension supplies the site directly.