For Windows Server computers, please also enable ConfigureOnPremisesAccountAutoSignIn policy and set it to 'SignInAndMakeDomainAccountNonRemovable' since you are using OnPrem accounts. Microsoft Edge will automatically sign in users that are on domain joined machines using their Active Directory accounts in this way.
For Workstation computers, normally the user gets automatically signed in with their Microsoft Entra account if they are AAD-joined/hybrid-joined. You may consider enabling NonRemovableProfileEnabled as well.
EDIT:
For Windows Server computers,
- If you need to use Edge as a published application, the only option is to continue using OneAuth as usual (no specific configuration is required) and instruct the impacted users that they need to complete sign in when launching Edge.
- If you are able to use Edge with a full desktop application, then enabling WAM support in Edge for Windows Server 2016 (if you are using it) is the best way to go. This is enabled by the policy WAMAuthBelowWin10RS3Enabled.
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Best Regards,
Shijie Li