This Wikipedia article seems to correctly define the real-world concept of administrative center as a capital city or other place, not a capitol building. Maybe you came to a different conclusion because of the image of an administrative building at the top of the article?
No, the main reason admin_centre is so diluted is that label is so poorly named. admin_centre is much more frequently misused for what should be label than for a capitol building, though that often happens too. Making matters worse, until 2020, the boundary relation documentation failed to clearly describe the label and admin_centre roles. In fact, label was documented as a rendering hack, even though it’s only ever been used that way in a vanishing minority of cases.
Even though admin_centre is a fine name for the role, I agree that it has been diluted to the point of uselessness. The few data consumers that support this role incorrectly equate it with label. Unfortunately, I’m pretty sure if we were to introduce a capital role, misuse for capitol buildings would become much more common. With both capital and capitol roles, mappers will likely typo and misread these roles all the time. Even native English speakers frequently confuse these distantly etymologically related words.
I’m a little hopeful that iD can soon help by providing a more intuitive, validated UI for relation roles in general. Practically speaking, that would be an important step for implementing any deprecation of a widely used role, since iD currently suggests the most common roles from taginfo.
I don’t recall bringing it up or drawing any conclusions, but I guess you mean one of these discussions?