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Inside Joomla System Extensions under Admin Tools Core package it says update key is missing

Um, Admin Tools Core is the free of charge version which never has a download ID. Downloads for it are free of charge.

Your screenshot shows the updates for Admin Tools Professional for Joomla having a Download ID.

What I understand is that you have two Update Sites records, one for Admin Tools Core and one for Admin Tools Professional. Select the checkbox next to the Admin Tools Core update site, and delete it.

This is a real gap in Joomla's extensions installer. When you install a newer version of an extension which has a different update site, or indeed when you upgrade from a free to a paid version / downgrade from a paid to a free version, Joomla does not always replace the extension's update site with the one from the package you installed last.

One solution is what I described, deleting the odd, old update site manually.

The other solution is to delete all update sites for the extension and click on the rebuild button in the Update Sites page. This goes through the extensions' XML manifest files cached by Joomla at installation time, filling in any update sites missing. This does install the latest version of each extension's update site. However, you will need to re-enter the Download Key for the extensions whose update sites you removed and had Joomla rebuild. That's why I only suggest it in truly hopeless use cases.

FYI, you must only ever uninstall the package extension. Never try to uninstall the subextensions (component, plugins) by themselves. The reasons are a bit technical and convoluted. It really all comes down to Joomla sometimes mishandling this to the extent that it loses the connection between the package and the subextensions, leading to problems with updates and false incompatibility reports when you update Joomla to a newer minor or major version. This is why I did not tell you to do that. The other two options I gave you, including rebuilding the update sites, are perfectly safe.

If you get messages about Admin Tools' plugins being incompatible with Joomal 6.3 in about 6 months, I will tell you what to do. If you do not see any such messages – which is very likely as the issue I described only happens under some specific conditions – you won't have to do anything. I am only mentioning that so that you understand why I did not ask you to do what you did, even though I know it is possible. I just know that a misunderstanding, deviation from the instructions, or a misclick can cause far worse problems than what it solves.