I cannot reproduce your issue. I installed a fresh copy of WordPress 6.9.4 and enabled the multisite feature. I even created two blogs in it with separate users. I then installed the same developer release of Admin Tools Professional (2.0.2-dev202603101521-rev19588b49) I linked you to.
I was able to see and use Admin Tools Professional just fine with the default Super Admin user.
IMPORTANT! Do note that the URL for Admin Tools' control panel page is /wp-admin/network/admin.php?page=admintoolswp%2Fadmintoolswp.php. The very important part is that the last bit MUST have the forward slash URL-escaped to %2F. If you do not do that, your web server might misinterpret that as an attempt to find a directory called admin.php?page=admintoolswp under /wp-admin/network which will, of course, result in a 404. Whether this happens depends on your server configuration, including your .htaccess content.
I then created a new user test in the network admin panel and gave it Super Admin privileges.
IMPORTANT! By default, even when using the Network Admin's Users page, WordPress creates a user for the main site, without any special privileges. To be able to use Admin Tools this user MUST be elevated to Super Admin, not just a regular Administrator. Super Admins can access the Network Admin Dashboard where Admin Tools and Akeeba Backup live. Regular Administrator can only access the content (posts, pages, widgets, config options, ...) of the main site. To do that, you need to edit the user after creating it and check the "Grant this user super admin privileges for the Network" box next to the Super Admin option.
I logged out from my regular Super Admin user and logged in with my new user test.
IMPORTANT! Do note that logging in with this user shows the main site's admin panel, NOT the Network Admin Dashboard. I had to go to the top menu, My Sites, Network Admin, Dashboard to see the Network Admin Dashboard.
In the Network Admin Dashboard I can see the Admin Tools sidebar icon. Clicking on it allows me to use Admin Tools Professional perfectly fine.
Do remember that both Akeeba Backup and Admin Tools will only appear in the Network Admin Dashboard which is only accessible to Super Admin users because they can be used to effectively escalate one's privileges to Super Admin, and have access to the totality of the information contained in your site. This level of access MUST be reserved for the highest privilege level available on your installation, i.e. Super Admin on WordPress multisite installations and Administrator on WordPress single-site installations. Anything else would be a privilege escalation and/or data leak vulnerability.
As far as I can tell, your problem appears to be that you are using the wrong URL with a user that does not have the correct privileges and you are looking for the plugin in the wrong dashboard. I am not blaming you. The WordPress multiuser experience is a mess. Having a main site dashboard and a network dashboard accessible by the same users with the same login URL is unconscionable.
Nicholas K. Dionysopoulos
Lead Developer and Director
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