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Admin Tools does not block or otherwise interact with Joomla's extensions updater.

The problem here is of a different nature. These icons are updated asynchronously. The browser sends AJAX requests to your server to request the status of the updates, overrides and so on. However, Joomla is trying to send all these requests at once. From your server's perspective, there is a barrage of requests coming from your IP address. From where the server is standing (it doesn't know tha nature of these requests, why or how they were initiated) it looks like a Denial of Service attack. Most servers will block some or all of these requests. The same problem existed in Joomla Update's pre–update checks for new minor and major versions of Joomla. I have contributed a fix for Joomla Update which will hopefully make it to Joomla 4.0 before version 4.1 is released but I have not tried to fix the status check quickicons (what you have a problem with) and I don't think the core contributors have understood yet the nature of the problem I just described to you.

In extremely rare cases you may have a different problem: you may have a $live_host set up in your configuration.php file which does not agree with the settings you have set up in the .htaccess Maker. For example, if your configuration.php file reads $live_host = 'http://example.com' and you have enabled HSTS and redirect non-www to www in the .htaccess Maker you have a problem: your site is NOT accessed through http:// (HSTS means it's always accessed by https://) and the non-www to www redirection means that your site is always accessed as www.example.com, not example.com without the www in front. This discrepancy will make AJAX requests to fail as well as fonts not load. Since I see the graphics of the icons which are loaded  through a font file (Fontawesome 5) I don't think that's the case which is why I am only mentioning this last. Do check your configuration.php file, though. If it has a value for $live_site unset it, i.e. make that line read public $live_site = '';