This is where the term "public" gets a bit sticky, as there are so many different definitions. For example, anything owned by the government is technically "publicly owned" but that does not mean accessible to the public, as a person is likely to find out if they try to enter certain military bases (particularly if they don't go through the one of the gates of entry). There are also areas like parks or cemeteries, where they are "publicly owned" and even "publicly accessible", though they may have hours they are open and closed, and being there while it is closed can get you arrested.
Courts are one of those weird cases where they are technically open to the public but with a lot of restrictions. For example, family court and grand jury proceedings are almost always closed to the public. There are other restrictions in terms of recording devices where just carrying a camera or some type of audio recording device can get you blocked or removed from the courthouse. Additionally, despite the fact that you have a Second Amendment right, you can't take your gun into most courthouses. What is more, something simple like deciding to go to the courthouse to eat your lunch while watching a court case is likely to get you removed (if not found in contempt of court). While the public is allowed, they are limited as far as rights, which to my understanding means they aren't "publicly accessible;" there are limitations to public access (no guns, no recording devices, no disruptions, etc).
So, again as I understand it and have read about it, because of the limitations of public access, administrative warrants are not allowed to be acted on in courts (at least as a general rule). In fact, this courthouse appears (per the reporting of the story) to have policies that prevent warrants in general from being served by law enforcement (I suspect there are exceptions for judge mandated arrests in their courtrooms), with a major reason for this rule is that acting on arrest warrants in the courthouse (even in a hallway) because it can cause interruptions and even safety issues with court proceedings -- and this could be doubly true if a person they are attempting to arrest is in a hallway and "runs" and attempts to go through various courtrooms in their attempt to escape law enforcement.