>>>Would Chief Rabbis have authority in their own right, without being poskim?!
No. A government appointed official is no more of an authority on Halacha than any yungerman who went through the sugyos. If even that. Unless of course he is independently a Talmid Chochom
It is very rare however that a government appointed rabbi is also a respected Talmid Chochom in areas of halacha. When was the last time you heard anyone mention a psak halacha from the chief rabbis of France, South Africa, England, Iran or Morocco?
On the other hand not dealing with their situations there is no way you are in the position to mach avek their behavior in a situation where there is definitely room for them to do so.
(Rabbi Herz mentioning bible critics is completely independent of his being chief rabbi of England. I was also told that he himself told people not to learn his commentary on chumash. He just felt that a time when bible critics held so much sway and influence a commentary on chumash written for a secular audience can’t ignore them. I disagree but that is not a England specific question like this is)