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Since we are not given the history of the sites in question, I will have to assume it is what Davide explained in your previous private ticket:

This happens because core Joomla tables are "dirty". Joomla finds some previous records and thinks it has to update instead of install (or viceversa). This usually happens when you have a very old site (from Joomla 3) and you updated it until now. Some leftovers build up and you might have this kind of issues.

The best thing to do, if this happens again, is to export, uninstall + install, import again.

I would also add the situation where you have manually deleted Admin Tools' files, without actually uninstalling the package extension from Extensions: Manage. This would indeed make Joomla throw the error you mentioned when trying to reinstall the package since it “knows” that Admin Tools' tables are present (based on the #__schema database table entry telling it so), but the tables are no longer present in the database. In this self-inflicted problem situation when it tries to execute the SQL to upgrade the Admin Tools tables (which are not actually present) a database error occurs.

There are other, more subtle, ways you can mess up Joomla to the point that uninstalling and/or reinstalling an extension ends up in a database error. Since you are reporting that the menu item has disappeared –which implies you have done something to your site's database or tables that you shouldn't have done– we gave you the steps to safely recover from this situation.

To be clear, the steps are:

• Export the Admin Tools settings (if possible).
• Uninstall the Admin Tools package extension. DO NOT try to uninstall just the component extension; THIS IS CRITICALLY IMPORTANT!
• Install the Admin Tools package.
• Import the Admin Tools settings.
• Remember to create the .htaccess afresh if you were using the .htaccess Maker.

Since you have a tendency of breaking your Admin Tools installation in the same way, I would recommend keeping an export of your Admin Tools settings for each one of your sites for the next time something breaks.

To be absolutely clear, what you are experiencing is NOT normal, and NOT something we have seen within any significant frequency. Back in 2021–2022 when thousands of sites were being upgraded from Joomla 3 to 4 we did see half a dozen sites with this issue in total (out of thousands!). However, we also knew why it had happened. These sites went back to the Joomla 2.5 days or earlier. Sites which went from Admin Tools 2.4 to any version past 2.7 without installing any of the intermediate versions would have that problem. These were very rare, as we expected. I honestly don't think that you have several sites with this kind of unique history on your hands, which is why I am suspecting that you or someone else with access to your site has maybe applied a sequence of actions that Joomla disagrees with.