Wang Hui：
“Anything but” means “definitely not; not...at all”. When someone or something is anything but ..., it means that the person or thing is definitely not... or he or it is the opposite of ... (a certain quality indicated by the adjective or a certain noun)
For example, (1) They said John was hypocritical, but when we talked to John, he was anything but hypocritical.
(2) David is an honest man; he is anything but a hypocrite.
Other examples of similar nature:
(3) That old bridge is anything but safe.
(4) Was the weather fine that day? - Anything but.
(5) Your sentence: “It should come as no surprise to most readers that 'objective' government studies are often anything but .” Here, if you repeat “objective”, it is easier to understand: “... anything but objective.”, meaning: “such studies are definitely not objective.”
Besides, this phrase can also mean “except” when it is followed by a noun or pronoun, as in:
(6) I can give you anything but my diary.
(7) She said she would do anything but this.
(6) means “Except my diary, you can have anything you want.”（除了我的日记之外，什么都可以给你。）
(7) means “Except this thing, she would do any other thing.”（除了这件事，她愿意做其他任何事。）