The problem that I see is that statues HONOR the person, either for who they are or for what they did to make lives better for others. A test should be that they are "...an edifice which has answered in any tolerable degree for ages the common purposes of society." (From your quote from Burke.

Some statues "honor" people who led a massive insurgency against the United States. There is no honor in that. You can say that this particular statue is of a "good" and honorable person, but who makes that determination? It is a really gray area. But why do we need statues to honor those who we want to honor? I'm not saying we should get rid of all statues, but we can maintain our connection to our ancestors without them.