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Part of the issue is that I believe you said that some of the systems no longer have the Domain Admins in the Administrators group.
That problem will not be easy to bypass unless you know the credentials of another account on the system that does have Admin rights.
One thing you can do is use the users own rights to put the Domain Admins back. If they took them out then they must have Admin rights.
During every logon put the Domain Admins back. After a while they will get tired of removing them.
If you don't know the Admin account or equivalent password then you will have to walk out and manually reset the Admin password and then setup security properly so it doesn't happen again.
If you know the credentials then WMI / ADSI will easily allow you to add an account from your workstation.
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