Smerel, the chief rabbi is NOT appointed by the government! He is appointed by the United Synagogue, which is the organization he is chief rabbi of. He is not chief rabbi of those Jews not affiliated with the US, such as the Chareidim, the Sefardim, and the various heretical sects. So the whole question of government appointment is irrelevant.
AAQ, the Anglican church is theologically very diverse. Within the Anglican church you have a split between “High Church” which is closer to Catholicism and “Low Church” which is far away from it. Historically there were bitter struggles between these factions. But in general, and especially at a “High Church” church such as Westminster Abbey, there’s much more of an avoda zara problem than, for instance, at a Reformed Church, or the Puritan churches of the 18th century.
And no, the difference goes well beyond statues. In fact statues are not really that much of a problem. Kosher they’re not, but comparatively speaking they’re small change. The one thing in a Catholic church that is absolutely 100% unmistakable avoda zara is the “consecrated Host”. Almost all protestants, including even most High Church Anglicans, reject the doctrine of transsubstantiation, and therefore the problem is smaller. And most protestants don’t even have the whole concept.
All other issues with Xianity are less important. It’s questionable whether worshiping J. is avoda zara at all. They mistakenly believe that he is HKBH, so when they worship him their intent is to worship HKBH, rather than a rival god. (The same is true with the Host, but that is a physical object. When you believe that a physical object is HKBH, and you bow down to it, I don’t see how that can be anything but classic avoda zara.)